Just a suggestion, you should offer some explanations on your notation so readers unfamiliar with, say, conditional probability understand, say, the English version of it; plus, it just makes everything clearer if it can be spelled out. Makes your blog redundant, but more accessible. Also, by reference set are you referring to the universe of discourse?

@bryangoodrich - 

Yes, reference set=universe of discourse

@Spoonwood - 

A subset B of a reference set X, does depend on its reference set in the sense that one can define the subset B in terms of a collection of indicator functions for its members with respect to the reference set X.

Just because you can make such a denotation, does that mean it is dependent? I'm not seeing where the semantical paradox arises. I assume you're trying to get at what is meant by (in)dependence. But independent of what? dependent how? are the first things that come to mind. You say you can express the subset through that indicator function, but what kind of dependence is that showing? is that the kind of (in)dependence being captured by the set theoretical denotations? Is it a relevant distinction to draw out for sets?

@bryangoodrich - 

[Just because you can make such a denotation, does that mean it is dependent?]

It depends on how you define 'dependence'. The semantical paradox arises, because we don't usually think we need to define 'dependence'.

[You say you can express the subset through that indicator function, but what kind of dependence is that showing?

Dependence on the reference set. The subset A depends on the reference set X for elements it may have. The subset A depends on the reference set X for how much the elements of A, a1, ..., an belong to the reference set X or whether or not they belong. This happens from the definition of X and the definition of A if we define them by words. If we define X as the set of prime numbers, and A as the set of odd prime numbers, then A depends on X for its possible set of numbers, since any odd prime number must also qualify as a prime number. A property z of a subset A depends on the fact that the superset X also has also that propety z, otherwise set A no longer qualifies as a subset of X. Also, suppose we roll a die. The probability of an event for the throw depends on the numbers on the side of the die, which here consists of the reference set.

[Is that the kind of (in)dependence being captured by the set theoretical denotations?]

No, not exactly. But, one might expect this. I think the distinction relevant here.

@Spoonwood - 

The problem with the kind of dependence you are trying to describe is that the link between the universe U and the subset S needs to be defined. You are trying to define it merely by the power of it being a subset. But that is like saying I depend on the universe because I exist in it. In a sense that is true, but that intuition is correct because of what we call a causal connection. Sets are abstract objects and do not admit of such properties. Likewise, probability of events assumes a kind of real application in which we can define a linkage based upon probability axioms and the kind of intuition they are capturing about causal relations. Unless we assign the probabilities arbitrarily, we obtain them from actual information (not saying it needs to, we can do it arbitrarily and still work it out mathematically, bit it doesn't infer anything about real information--it's not applicable). The independence is trying to capture the independence of real events, as applied to real information. But set theory alone does not have this characteristic. There is nothing in the defining of sets that makes S depend on U since their defining implies every logical consequence that comes with it. For instance, defining an algebraic structure doesn't make the conclusion I draw about some property I discover later separate from when I defined the structure. They go hand-in-hand. It wouldn't make much sense to say that property depended on the algebraic structure. Likewise, S and U are always independent in that sense because there is no kind of dependence to be captured as we think about in the world. But what you are doing is providing a way to define S in terms of U to make them dependent by the indicator function. It just seems ad hoc, though, to throw in this indicator function out there to produce a kind of dependence, because that is what seems to occur. S is defined through I(m) off of U by relating S to U through I(m) for m being the members indicated in S from U. But off of ZFC there is no reason for the indicator function. S is definable without it and would be independent of U. Of course, we might say that our universe without I(m) is different from the defined universe of discourse with it because one has a concept of set dependability and the other does not. But if we make such a distinction we are no longer dealing with basic set theory (ZFC) but something that includes an indicator function. Of course, ZFC doesn't include large cardinals, but we can just as well add them and see what we can conclude (e.g., the continuum hypothesis), but then we are not dealing with ZFC as we know it. ZFC doesn't include a concept of probability, but based on a borel measure or something we can use sets and set theoretical properties to talk about measuring probability for applications, among other things (i.e., some find it meaningful for its own theoretical purposes, even if we don't apply it to, say, statistics). What is the purpose of the indicator function and the consequences of pulling it out?

@bryangoodrich - 

[You are trying to define it merely by the power of it being a subset. But that is like saying I depend on the universe because I exist in it. ]

I don't see the latter statement as a problem whatsoever. In fact, I suspect I may have said it in some comment somewhere on someone's blog before.

[Likewise, probability of events assumes a kind of real application in which we can define a linkage based upon probability axioms and the kind of intuition they are capturing about causal relations.]

I disagree. Let's say we have the sample space or universe of discourse {1, 2, 3, 4, 5}=X with the event {3, 4, 5}=A. Given no weighting of elements, what's the probability of A? 3/5. What type of causal relation happens here? I certainly don't see any (maybe I misunderstood you here). Also, we *can* define conditional probability and therefore probability in general purely in terms a subsethood relation. For the above, we just ask "what degree of subsethood does the set X have in A? It's 3/5 and that number (not 3/5, but the degree of subsethood of X in A) equals the probability in general, as Kosko shows in his fuzziness vs. probability paper.

[But what you are doing is providing a way to define S in terms of U to make them dependent by the indicator function. It just seems ad hoc, though, to throw in this indicator function out there to produce a kind of dependence, because that is what seems to occur.]

The indicator function doesn't get thrown in there. Given a property for a subset A, the indicator function defines which elements a1, ..., an belong to the subset A which satisfy that property. If we talk about the subset A of odd prime numbers, the indicator function defines which elements of the set X of prime numbers (or natural numbers or real numbers or any superset of odd prime numbers) belong to A. For perhaps a bettter example suppose we talk about the subset which satisifies the equation 4n+3 (with n belonging to N+) for the universe of discourse of {1, 3, 5, 7, 9, 11, 13, 15, 17}. In such a case, the indicator function tells us that we have the subset {(1, 0), (3, 0), (5,0), (7, 0), (9, 0), (11, 1), (13, 0), (15, 1), (17, 0)}. Of course we can just write (11, 15}, but that's just a shorthand for the above. For any set Z (so far as I can tell), there exists an indicator function there already, it just doesn't get explicitly written.

[But off of ZFC there is no reason for the indicator function. S is definable without it and would be independent of U.]

Alright, we agree that there exists a membership relation for all sets, correct? And I admit that I don't know ZFC all too well, but it does need a membership relation for all elements of a set? We have a single primitive binary relation: membership, correct? For any set Z, with elements z1, ..., zn we have z1 belonging to Z or not belonging to Z. The indicator function just assigns two distinct numbers "1" and "0" which correspond to "belonging" and "not belonging". In principle (as I understand it), it could assign .5 to belonging and 1.5 to not belonging. So, the indicator function just gives us another representation of talking about "belonging" and "not belonging". Consequently, it seems that if you've defined the membership relation, then you've defined the indicator function other then the matter of what numbers you assign for "belonging" and "not belonging". In other words, the indicator function seems a necessary result of having defined the membership relation.

I also don't get injecting ZFC here in general, since I didn't bring it up nor make those axioms my reference. I could have brought up Von Neuman-Godel-Bernays axioms or non-well-founded sets or even "naive" set theory and foregone a modern axiom characterization altogether.

You said earlier "But what you are doing is providing a way to define S in terms of U to make them dependent by the indicator function." S works as dependent on U in terms of the membership relation. The indicator function reiterates this.

[But if we make such a distinction we are no longer dealing with basic set theory (ZFC)...]

I disagree with the implicit propostion here that "basic set theory" is ZFC. ZFC comes as an approach to an axiomatic set theory. There exist other axiomatic approaches. In addition to that, an un-axiomatic approach to crisp set theory comes as even more basic than an axiomatic approach. This doesn't mean that ZFC lacks value or that any other approach to an axiomatic treatmenet exists. But, it does mean that we would preferably not assume ZFC when someone discusses sets until they indicate ZFC.

[but something that includes an indicator function.]

I contend that if you have a membership relation, then you have an indicator function. The choice of numbers for the indicator function makes no difference for the theory itself (for the development of the theory by humans, that's a different story)... A={1, 3} still qualifies as a subset of the reference set B={1, 3, 5} if we assign .5 to belonging and 1.5 to not belonging, where in such a case A={(1, .5), (3, .5), (5, 1.5)} B={(1, .5), (3, .5), (5, .5)} where in such a case 1.5 precedes .5, so (1, .5) precedes (1, .5), (3, .5) precedes (3, .5) and (5, 1.5) precedes (5, .5) and consequently A qualifies as a subset of B. So, I think that any axioms for set theory which have a membershp relation, already have an indicator function. The assignment of numbers 0 and 1 specifically just makes things computationally easier to deal with (and in other ways also).

Maybe I haven't talked about "the indicator function" as conventionally defined, but I feel rather sure I've talked about some concept so close to that one that a distinction seems rather picky.

[What is the purpose of the indicator function and the consequences of pulling it out?]

To me that's like asking "what's the purpose of the membership realtion and the consequences of pulling it out?" The indicator function tells us all sorts of things really, if we care to look. Using the indicator function, we can readily see that basic equations of crisp set-theory such as De Morgan's Law works out as isomorphic with that of Boolean Algebra. We can also write every set in terms of a bit vector (given that we order a set in a particular way) for our reference set. For instance, say we have {2, ... 10}. For the subset A of odd numbers, we could just write 010101010 for the subset A. We can also graph sets on Cartesian-Oresme coordinates using the indicator function (if we don't get given a reference set, we can just take the set in question as its own reference set, since every set qualifies as a subset of itself).

@Spoonwood - 

I will provide more details later, but the indicator function, or any function for that matter, is not part of the sets or their relations. You imposing that on there provides something, but it does not come part in parcel with sets. Once you add the indicator function, you're dealing with a whole other system of analysis from set theory itself. That is my point in the fact you find no indicator function involved, nor implied, by ZFC. Certainly there are other approaches to sets, but the indicator function is neither inherent nor required. By adding it you are providing a whole other set of analysis, just like adding the Borel measure to do probability analysis makes probability theory a whole other analysis from set theory itself.

As for probability, your example of the subset is completely arbitrary. The point was that independence is trying to capture real-world independence, that what effects some event A does not effect B, and this is expressed, if our measure properly applies to reality, by the real probabilities we can observe. If we're dealing with these abstract tools without such information, then it is all arbitrary. They are good for classroom examples to understand how they work, but seeing them in application is the point of their use. Not that there is anything wrong with learning probability measures or integration measures in their own right. Calculus and sets are fun to math nerds, but they have developed these tools, in part, to explain real events. The point was that independence is trying to capture X. You are trying to capture Y with talking about it in another fashion. But using the universe and an indicator function are not inherent in sets, nor need to even be comparable to that captured by probability. Dependence and independence, anyway, requires there to be a relation that we define as necessary. The real-world we take to have certain necessary (causal) relations. That is what independence is trying to capture, in part. But there is no necessary relation described by the indicator function unless you can prove it is inherent in sets. Is it?

Point in case, there is no "membership relationship." A member is not related to the set as if they are two separate things. The set IS the members. I am going farther than I want at this point (my resources are at home), but that is the general idea. I like what you say about what the indicator function can offer, but that is like trying to show any kind of measure theoretical property that can be spelled out over sets is inherent in sets. It isn't. That application, that measure, is a whole other analysis, taking other things to be assumed for it to work (like probability axioms, etc.). I would like to see more examples of what can be done with this indicator function. It applies directly on a set as you can spell out, for instance, a bit sequence to describe the odd numbers. That is especially useful in computability, too. Do you have that proof where "we can readily see that basic equations of crisp set-theory such as De Morgan's Law works out as isomorphic with that of Boolean Algebra"?

But my argument still remains, if the indicator function is not just "throw in there" then where does it come from? How is it inherent in sets? If so, ZFC is taken to be our most sophisticated representation of sets given the least amount of axioms to make it work, and it is not implicit in ZFC at all. You'd have to show, I would think, either it can be drawn out from ZFC or from some naive set theoretical concepts, inherent in our concept of sets and memberships themselves. But as I said, it is not apparent to me unless you look at sets separate from their members, but that would be like saying the number 4 isn't a member of the natural numbers, integers, rationals, etc. until it is indicated as such and made a member. That is not set theory. The number 4 is part in parcel with the denotation of those sets. I ask this to consider, are their separate indicator functions for 4 in N, versus 4 in Z, versus 4 in R, versus 4 in C? Is there a difference in these "4s" or are they the same one? Does it depend on a universe of discourse? Do set denotations need that universe?

@bryangoodrich - 

[The point was that independence is trying to capture real-world independence, that what effects some event A does not effect B, and this is expressed, if our measure properly applies to reality, by the real probabilities we can observe.]

I don't see us observing probabilities like we observe a tree. What in the world do you mean by saying that we can observe real probabilities? We calculate probabilities. We can calculate any probability P(A) as a conditional probability P(A)=P(A|X) since the reference set X comes as the sure event. We can do this by calculating n(A^X)/n(X) which also equals S(X, A). The subsethood measure S(Y, Z) doesn't get assumed into to existence as does the more conventional notion of conditional probability P(Z|Y). One can derive the subsethood measure S(Y, Z) from more basic priniciples. To see how to do that, you'll need to consult Kosko's paper Fuzziness vx. Probability .

I can see us inferring independence or dependence from real-world events. Maybe that's what you mean to say?

[But using the universe and an indicator function are not inherent in sets...]

Using the universe and using the membershp relation come as inherent in sets.

[But there is no necessary relation described by the indicator function unless you can prove it is inherent in sets. Is it?]

Membership comes as inherent in sets. Since membership determines the behavior of the indicator function, this means we can write every set in terms of its membership function, since you can write every set in terms of its members. That seems necessary.

[A member is not related to the set as if they are two separate things.]

Actually, a member x of a set A belongs to that set A. Since x belongs to A, it qulaifies as a member of the set A. It has to, otherwise it ceases to exist as a member. The set A also x as one of its members. They work out as separate in that a member qualifies as a single object, while a set qualifies as a set of objects.

[The set IS the members.]

I have a feeling you didn't mean this, but just in case you did, I respond... No, the set consists of the union of its members. One can describe the set of odd numbers less than 10 as {(2n+1): 0<n<5, n belonging to the counting numbers}. One can't describe the member 3 of this set as the set. One can't define 3 in terms of the property used for the set above.

[Do you have that proof where "we can readily see that basic equations of crisp set-theory such as De Morgan's Law works out as isomorphic with that of Boolean Algebra"?]

Maybe this part doesn't work out so well, but you can write all proofs for the equations of set theory by using the indicator function. Actually, I probably don't exactly mean Boolean Algebra, but something like it. Still. I'll quote Buckley and Eslami from their An Introduction to Fuzzy Logic and Fuzzy Sets . I'll have to adjust their notation.

"Let us show how these [the basic set theory equations) may be proven using membership functions. Let us first show the De Morgan identity c(A^B)=c(A) v c(B) in the equation (3.13) [it's the same equation]. Let C=c(A^B) and D=c(A) v c(B) and we show that C(x)=D(x) for all x in X. Now C(x)=1 if (A^B)(x)=0 and C(x)=0 for (A^B)(X)=1. So we see
1, if A(x) or B(x)=0
C(x)= 0, if A(x) or B(x)=1 3.18
Also, D(x)=0 [they mistakenly wrote 1] if c(A(x)) or c(B(X))=1 and D(x)=0 for c(A(x)) and c(B(x)=1 [they mistakenly wrote 0]. Hence
1, if A(x) or B(x)=0
D(x)= 0, if A(x) and B(x)=1 3.19
From equations 3.18 and 3.19, C(x)=D(x), for all x in X, and this De Morgan law holds."

The text continues:

"For another proof let us show the absorption law A^(AvB)=A of equation (3.17) [like above, it's the same equation]. Let C=A^(AvB). Then C(x)=1 if A(x)=1 and (A v B)(x)=1 and C(x)=0 if A(x)=0 or (AvB)(x)=0. Therefore, C(X)=1 if A(x)=1 and (A(x)=1 or B(x)=1) and C(x)=0 if A(x)=0 or (A(x)=0 and B(x)=0). Hence, C(x)=1 if A(x)=1 and C(x)=0 when A(x)=0 and C(x)=A(x) for all x."

Using characteristic functions on {0, 1} we can also put union and intersection within the more general framework of t and s-norms (or t-conorms) given that we have an ordering for '0' and '1' so that we have a lattice. http://en.wikipedia.org/wiki/Triangular_norm Mathworld says this "Given a subset A of a larger set, the characteristic function chi_A is defined to be identically one on A, and is zero elsewhere. Characteristic functions are sometimes denoted using the so-called Iverson bracket, and can be useful descriptive devices since it is easier to say, for example, "the characteristic function of the primes" rather than repeating a given definition. A characteristic function is a special case of a simple function." http://mathworld.wolfram.com/CharacteristicFunction.html

[But my argument still remains, if the indicator function is not just "throw in there" then where does it come from? How is it inherent in sets?]

It comes from the notion of membership. If x belongs to X, then the characteristic function of x in the subset X (remember, *every* set qualifies as a subset of at least one set... at least itself) equals '1'. If x does not belong to X, then the (perhaps I should say *a* characteristic function, since perhaps I've generalized the concept) characteristic function of x in the subset X equals '0'. The indicator function comes from instantiation of the general notion of membership with respect to particular members. It comes from the membership relation.

Let Ca(x) stand for the characteristic (or indicator) function of x in the subset a. Instead of writing
0 if x ~e a
Ca(x)= 1 if x e a

Perhaps I should write
z if x ~e a
Ca(x)= y if x e a

Where z and y indicate distinct items and z logically precedes y (although, z does not necessarily need to come as less than y).

[But as I said, it is not apparent to me unless you look at sets separate from their members...]

You can look at sets separate from their members. Consider the set of objects which have a volume greater than your laptop computer. You can say certain things about this set without knowing its members. In fact, looking at this set in terms of all its members simply doesn't work out as practical, so in some sense you have to look at that set separate from (some of) its members. We look at infinite sets all the time separate from its members. For instance, we can prove that for all members o of the set of the odd numbers O and for all members e of the set of even numbers E, the product of e and o yields a member e of the set of even numbers E without looking at any member of the sets e and o. We only need the properties e=2n and o=2n+1.

We can also look at certain conjectures about the infinite set of primes through looking at just finite subsets... meaning we don't look at all the members of the set of primes. Which conjectures? At least the ones we can disprove through finite examples, such as that all fermat numbers Fn=2^2^n+1, n e N qualify as prime. This tells us something about the infinite set of prime numbers by only looking at some of its members.

[I ask this to consider, are their separate indicator functions for 4 in N, versus 4 in Z, versus 4 in R, versus 4 in C?]

Actually, yes, even though the indicator function gives the same result. We have
CN(4), CZ(4), CR(4), CC(4), so each indicator function acts with respect to a different subset.

[Is there a difference in these "4s" or are they the same one?]

The first 4 comes as a member of only N. The second one comes as a member of Z (and also of N), and so on. If we have N as our reference set or universe of discourse, then we can't say that 4 belongs to C without discoursing about something outside of our universe of discourse. In other words, if we have N as our universe of discourse and start talking about a more general status of 4, we've changed the subject. Maybe we should do that more often, but preferably we at least recognize we've changed the subject when we do so.

[Is there a difference in these "4s" or are they the same one?]

In terms of how you can write them (or equivalent expressions), YES. You can write 4/1 for 4 in Q or R or C, but you can't write in N or Z. You can write 4+0i in C, but you can't write it in N, Z, Q, or R, since 'i' doesn't exist in N, Z, Q, or R. This affects all sorts of things which you can logically say about 4. It affects how you describe 4. Does it matter with respect to what "4 is in itself."? But, that's the whole thing, what in the world do you mean by "4 is in itself?" I certainly don't know. So, you'll have to describe to me what you mean by "4", and your descriptions in part depend on your actual allowable terms of discourse for "4" (which might not work out as a clear defined set). I could also throw the question back to you and say "does there exist a separate membership function for 4 in n, versus 4 in Z, versus 4 in R, versus 4 in C, versus 4 in F (the set of fuzzy numbers)?" In this case, the answer comes out an even more resounding "yes", since 4 in F can take membership values between 0 and 1 as well as 1 for 4. Note that every indicator function qualifies as a membership function, so I've only generalized the question.

[Does it depend on a universe of discourse?]

If no universe of discourse gets specified and we start talking about a set Z, then by default one can treat Z as the universe of discourse, since every set works out as a subset of itself (to some non-zero degree). Every set depends on itself in this way.

[Do set denotations need that universe?]

Every set denotation needs at least its own set as a possible universe of discourse.

@Spoonwood - 

I don't see us observing probabilities like we observe a tree.

That's fine, I didn't mean it like that. It's common practice to refer to known probabilities as observed probabilities, because you observe the phenomena, record and calculate frequencies, say, to determine them. This is opposed to expected probabilities, which are calculated and unobserved or unobservable since the future is not now. There is nothing strange in my usage, as any statistics text book refers to them as observed values. It's the nomenclature when doing a chi-squared goodness-of-fit test, for instance (i.e., the sum of (O-E)^2/E).

I meant independence is a real-world phenomena. What independence in probability theory is trying to capture is that aspect of the phenomena. It is a part of the probability measure involved in the application. We can determine it by looking at observed values. This goes hand-in-hand with the fact probability is a real-world kind of measure. There is no probability that just exists in mathematics. That would be like trying to talk about causality in mathematics. It wouldn't make sense. Thus, independence as related to probability as expressed through probability theory (an extension from set theory) is attempting to capture (model) real-world phenomena.


Now, you seem to be obfuscating the concept of sets and members. To demonstrate that, we shall use your example.

I have a feeling you didn't mean this, but just in case you did, I respond... No, the set consists of the union of its members. One can describe the set of odd numbers less than 10 as {(2n+1): 0<n<5, n belonging to the counting numbers}. One can't describe the member 3 of this set as the set. One can't define 3 in terms of the property used for the set above.

What is a union of members? Union is an operation over sets, unless you're going to say the member is a set (and yes, we can express them as sets. Hell we can express them using just the empty set). But I never said the set is any one member. Saying "One can't describe the member 3 of this set as the set" makes no sense. I said the set IS the memberS. To quote A. H. Lightstone Symbolic Logic and the Real Number System (New York: Harper & Row, 1965), "a set is defined iff we can assert of each object either that the object is a member of the set or that the object is not a member of the set" (p. 45). The fact that I can name (i.e, denote) the set by its members is a clear enough example of the fact that a set IS its members. A={1,2,3,4,5} is saying 1,2,3,4,5εA. Of course, it makes more sense to define the universe of discourse for any investigation, since one can wonder what 1,2,3,4,5 mean, but that is irrelevant to the fact the set A is its members. A is not separate from them. A, in fact, is nothing more than a higher-order property (from first-order logic) denoting and expressing a specific kind of relation over the universe of discourse--namely, that the elements mentioned are what we are calling A.

Every set denotation needs at least its own set as a possible universe of discourse.

They do not really need it, but there's nothing no meaning behind the symbols until it is spelled out. I can say %={~,#,J,9) and there is no confusion that what I am referring to is the naming of a set % by the members ~,#,J,9. I don't need to say where they belong. Just like I did not need to say 1,...,5 were from the natural numbers. We would most likely assume such a regular connection. Our statements, our denotation, can be that arbitrary though. We can define all sorts of relations among the denotations, but there's nothing meaningful until the domain and symbols are defined (with some sort of signature). In E.C. Wallace and S.F. West Roads to Geometry 2nd ed. (New Jersey: Prentice-Hall, 1998), the authors demonstrate this fact when discussing the axiomatic system (chapter 1.2, p. 8). You can spell out arbitrary axioms (relations) and even derive other statements (relations) from how the axioms interact with each other. But nothing meaningful needs to be derived until you define the symbols and then it may be the case that some of the theorems derived from these axioms in general may not apply. As the author's say, "Since these terms are truly undefined, they have no inherent meaning, and each reader may choose to interpret them in his or her own way. By giving each undefined term in a system a particular meaning, we create an interpretation of that [axiomatic] system. If, for a given interpretation of a system, all the axioms are "correct" statements, we call the interpretation a model" (p. 10). The sets can contain undefined terms (since the set itself is a higher-order relation over, say, first-order individuals). Spelling out a universe of discourse breathes life into these symbols, so to speak, but we don't need that universe spell out. I can still construct a well-defined set however (just like one can construct a wff for an axiom of some system, and may or may not be able to derive a model for it under some given interpretation, which would include the universe).

You suggest "by default one can treat Z as the universe of discourse" for some set Z, but that doesn't do anything. What meaning does it provide? Even if I looked at % as the universe of discourse, that doesn't change anything about it. In fact, a universe of discourse is nothing but the set of all designated objects under some investigation (Lighthouse 1965, p. 46). If all that has been specified is % and its members, then by default it is the universe of discourse. It is only customary, "for the purposes of a particular mathematical investigation...to designate in advance the objects under discussion" (ibid.). My question was specific to the fact sets do not need that universe of discourse. There is no substantial relationship inherent in it. The extensions are separate from the intensions. In other words, denotations are separate from semantics.


YES. You can write 4/1 for 4 in Q or R or C, but you can't write in N or Z. You can write 4+0i in C, but you can't write it in N, Z, Q, or R, since 'i' doesn't exist in N, Z, Q, or R. This affects all sorts of things which you can logically say about 4.

Actually it is rather an open problem in mathematics. There is absolutely no difference in denoting 4 as 4+0i, other than the fact you are making it explicit that we are dealing with an object from the complex numbers. But 4 is 4+0i. 4 also is 4/1. Just like 4 is 2+2 and 1+3 and 5-1 or 9+3 in Z_8. Once again, 4 is an arbitrary denotation unless specified. Specifying an object as, say, 4+0i limits the domains it can belong to since that denotation, we know, only exists in the complex numbers.

I could also throw the question back to you and say "does there exist a separate membership function for 4 in n, versus 4 in Z, versus 4 in R, versus 4 in C, versus 4 in F (the set of fuzzy numbers)?"

Now here is where you continue to obscure things. There is no inherent membership function over sets, at least in our common set theoretical methods (e.g., ZFC). Talking about the number 4 in different number systems does not admit of some kind of membership function, 4 just is or is not a member of that set since the set IS the members (see above). You can certainly specify a separate indicator function for each number system to say something about 4 in that universe of discourse, but that does not follow from, say, the axiomatic approach under ZFC. So where does the axiom, or something else, come into play that spells out an indicator function? I brought up the example about large cardinals because your response is like saying "we can solve the ambiguity problem of this number by the fact it has a different indicator function in each number system" but that is like saying "we can solve the continuum hypothesis by adding large cardinals to our system." That is fine, but we are no longer dealing with ZFC. So what is this theory you are appealing to? ZFC+{indicator functions}? Fuzzy set theory? What justifies its use? But, of course, that is not the case as you see it. You said the indicator function is inherent in sets! It isn't there under naive set theory nor our standard set theory (ZF or ZFC). So where does it inherent from? You might get away with saying it is inherent in our concept of sets that naive set theory nor ZFC couldn't capture to justify why we should include it, but that is a whole other discussion itself. It still remains the case that sets as we standardly approach them do not come with inherent indicator functions. It is like saying sets come with an inherent measure or an inherent probability function. It doesn't make any sense since those things require more to be spelled out. In fact, it goes along with defining the algebraic system in question that is being used, and that is a bit far removed from the sets or set theory itself (which deals more with foundations).

Now, what you say about separating members of sets is true, to some extent, but you've still convoluted the idea of sets. Simply because I can have an infinite set and not spell out each of its members doesn't change the fact we are still simply giving a denotation. In that case, we are naming it by a property as opposed to its members. But the set itself is still the members it obtains. Furthermore, to gain the usage of operations as you used requires us to define more than merely the sets. That is a like mistake you seem to be making with the indicator function. I'm not saying there's anything wrong with it or using it. My point is that it requires more to be spelled out about the investigation we are making. If we're just dealing with sets, then no, it is not present. If we define a certain universe and algebraic system, for instance, then we can start doing a whole lot more. We might even be able to add borel measures and talk about probability spaces, among other things. Of course, the indicator function provides something more foundational, but consider the fact you define a subset {1,2} of {1,2,3} by {(1,1),(2,1),(3,0)} when using the indicator function. This kind of usage demands the former to be in some relation with the latter. But when I define A={1,2} and B={1,2,3}, we only get relations between A and B because of the members. Our universe of discourse, as denoted, has to contain at least the three names 1,2,3. But let 3=2. Then A=B and (3,0) is wrong. Of course, the indicator function would have to change that fact as the facts have just changed, but my point is that as presented you are imposing a relation of A from B because B becomes the universe and A is in relation to it. But on what grounds do we say A is related to B like that? What defines a relation between sets? The sets are only related, by say the solutions to unions or intersections by power of the members themselves. A union B equals B because the members are what they are. A intersect B is A because as defined A is a subset of B. But we do not say A is a subset of B therefore its members are derived from B or something. That wouldn't make sense at all. But that is exactly the kind of thing you seem to be saying by what you said above about separating sets from their members.

Membership comes as inherent in sets. Since membership determines the behavior of the indicator function, this means we can write every set in terms of its membership function, since you can write every set in terms of its members. That seems necessary.

And in a way you are putting the cart before the horse. If all you have are members and an indicator function, for instance, then we can say we produce sets through the function acting over the members. But then where does the indicator function gets its information about how to place the members? If it gets them from the sets, then we had sets all along because sets are the members. There is no transition from members into sets. We just have members and we are drawing imaginary lines around them to enclose them and say "there, that's the set." Nothing was done other than provide a name, relating the members in such a way that they obtain the property of being a part of that set (in this case, the property is merely being in the set). Of course, we cannot draw such a line over an infinite span, but then again, I can, e.g., Z={...,-2,-1,0,1,2,...}. Of course, this assumes the pattern is apparent and constant, which it should be as defined (i.e., there shouldn't be any ambiguity in that notation). But you say membership determines the behavior of the function. But there is no behavior involved between sets and members! So if the function is inherent in sets and members, then why does it have behavior and sets and members do not? There certainly is a "behavior" we might say over me "drawing" that line which makes the set, but that is not contained it set theory. Maybe it's a lower-order event or some meta-theory thing, but it certainly is not present in set theory (e.g., ZFC) alone, as I expressed earlier.

Thanks for the example. There's a google books preview so I can see it in its original notation. You might want to spell out in the blog what the indicator (characteristic) function actually is. Particularly, it is a mapping in regard to a subset A of the set X onto Z_2 (yes, it does obtain boolean properties, though it is not a boolean algebra since it is not, as defined, an algebraic system--see the Lighthouse book, chapter 3 for more on that. Sells privately on amazon for 12 bucks. Good reference book since it covers logic, set theory, algebraic system and methods, number systems, real number systems (and some limit stuff), and even an appendix on the complex number system, all in under 300 pages in a small package!). I don't know where you gathered the idea that the characteristic function is inherent in sets. Any quotes or did you derive that one yourself? I didn't find it in the Buckley book. As described in Buckley, it just explains a transformation we can do over a subset (and it is defined specifically for each subset) in some universe (the super-set, we might say) that describes the membership more simply (binary or boolean). It just gives us a mapping to talk about membership in a numeric (Z_2) way to simplify these relations. Of course, once we change the mapping to [0,1] we allow ourselves to use a similar function to give us degrees of membership. Of course, then we're not dealing with standard set theory anymore (the mapping is different). Nevertheless, nothing is apparent about the characteristic function being "inherent" in set theory. The function is defined on sets, it transforms membership into something numeric, but the mapping itself is not inherent. We made it up. The property being transformed already existed and we just changed it to something usable in other contexts.

@bryangoodrich - 

[There is no probability that just exists in mathematics.]

Huh? Suppose we have the empty set as its own reference set. What's the probability of getting the empty set for any "experiment" P("0")? 1. Here we have a probabiilty which exists just inside mathematics.

You might feel tempted to point out that I've now violated some axiom of probability theory. However, NO axiom of probability theory says that P("0")=0. We have a theorem which says P(x)+P(c(x))=1. We do have an axiom that says P(X)=1, but that's fine with the above, so P(c(x))=0 granted that c(x) exists. But, here "0" consists of the entire space possible, so no complement of c(x) exists, not even the empty set (I admit this sort of reasoning contentious here).

If you don't like this example, then suppose we have the universal set X as its own reference set. What's the probability of getting the universal set X? 1. This probability exists purely inside mathematics.

[That would be like trying to talk about causality in mathematics. It wouldn't make sense.]

Causality works out as massively more complicated of a topic and basically has physical objects as its reference to begin with. Probability doesn't necessarily refer to physical objects.

[Thus, independence as related to probability as expressed through probability theory (an extension from set theory) is attempting to capture (model) real-world phenomena.]

Probabilistic independence exists for purely mathematical phenomenona. The empty set, with itself as its own reference set, works out as probabilistically independent of itself since P(0^0)=P(0)*P(0).

[I said the set IS the memberS.]

In which case the members are the set (symmetry of "be" verbs). So, we need members for a set. If the members are the set and we have no members, then it follows that we have no set. But, this doesn't hold. The empty set has no members and we still have a set. The empty set still has content such as A^0=0 and Av0=A for all A without members. A mere lack of members (whatever this means), without a set over it, has no content whatsoever. The empty set has content without members.

[To quote A. H. Lightstone Symbolic Logic and the Real Number System (New York: Harper & Row, 1965), "a set is defined iff we can assert of each object either that the object is a member of the set or that the object is not a member of the set"]

This works out as subtly different than the above statement, since it allows for the empty set to qualify as a set. Of course, the definition itself qualifies as only applicable to crisp sets.

[The fact that I can name (i.e, denote) the set by its members is a clear enough example of the fact that a set IS its members.]

You can't name members for the empty set. If you do, you don't have the empty set.

[A={1,2,3,4,5} is saying 1,2,3,4,5εA.]

Sure, and B={} is saying... that didn't work... I can't name members which belong to A!

[I can still construct a well-defined set however...]

You simply don't need a well-defined set to talk mathematically. Properly fuzzy sets (those whose membership functions lie in (0, 1)) DO NOT work out as well-defined since you can't tell if an object either belongs or does not belong to the fuzzy set. We can STILL talk about properties for union and intersection of those sets and prove theorems about them.

[My question was specific to the fact sets do not need that universe of discourse.]

You say this, but consider the following. Suppose we define a set
{x: x=4-6}. You say we don't need to specify the universe of discourse for sets in general, and I will instantiate your statement by applying it to this set (you'll need to clarify your meaning if I've hereby misinterpreted it). Alright, let's get down the nitty-gritty. Since sets don't need a universe of discourse, the set {x: x=4-6} doesn't need a universe of discourse. It's one set, free of the context of the universe of discourse. So, when it does appear within a particular universe of discourse it will come out the same, since it's always the same set. However, in N {x: x=4-6}={} since no '-2' exists, while in R {x: x=4-6}={-2}. So, the set, considers as context-free meaning without a universe of discourse, does NOT appear the same when instantiated all universes of discourse. So, the set {x: x=4-6} does NOT equal itself across different universes of discourse. But, sets equal themselves. Consequently, we can't practically accept that sets don't need a universe of discourse... they do... otherwise we'll end up with a massive contradiction as the one above.

[Actually it is rather an open problem in mathematics. There is absolutely no difference in denoting 4 as 4+0i, other than the fact you are making it explicit that we are dealing with an object from the complex numbers.]

Suppose I specify my universe of discourse as R. How in the world does i make sense within my universe of discourse?

[But 4 is 4+0i. 4 also is 4/1.]

4 equals 4+0i in the set of complex numbers. 4 equals 4/1 in the rational numbers. But, if we only have N as our universe of discourse, what does {4/3} equal? It equals the empty set, since 4/3 does NOT exist within our universe of discourse.

[Just like 4 is 2+2]

Consider our universe of discourse as {0, 1, 2, 3}. What does (2+2) equal? Again, ().

[Specifying an object as, say, 4+0i limits the domains it can belong to since that denotation, we know, only exists in the complex numbers.]

No, it doesn't. We can first talk about the domain of discourse or simply domain Z. We can then talk about {4+0i}. Well, i doesn't fall within our domain, so {4+0i}={4}.

[Talking about the number 4 in different number systems does not admit of some kind of membership function, 4 just is or is not a member of that set since the set IS the members (see above).]

No. If we have the fuzzy number 5=(3, 5, 7) with degree of membership (dom) of 3 equal to 0, dom(5)=1, dom(7)=0, and dom(4)=.5 we don't have 4 as a member nor a non-member of 5. 4 has a degree of membership in 5 and a non-degree of membership in 5. When you extend the question to F, as I did above, this happens.

[So where does the axiom, or something else, come into play that spells out an indicator function?]

The indicator function (for crisp sets) merely reiterates what the membership relation has already said in a different way. If you've specified the members, you've specified the values of the indicator function.

[I brought up the example about large cardinals because your response is like saying "we can solve the ambiguity problem of this number by the fact it has a different indicator function in each number system"]

I don't see this at all. I don't see where I brought ambiguity or addressed any sort of "ambiguity problem". Please stick to what I wrote and try to directly derive statements from what I wrote and please reference the statement itself that I wrote. Otherwise, you might create strawmen here. I didn't really intend to touch on large cardinals in my statements, since I didn't mean to address ZFC. In fact, I've indicated that I didn't necessarily mean ZFC when I started talking about sets.

[You said the indicator function is inherent in sets!]

Where exactly did I say this again? My original statement says "For instance, for {1, 3} we can define {1, 3} in terms of an indicator function I(xn) which here yields I(1)=1, I(3)=1, I(5)=0." And this holds, we can write any set in terms of its reference set (I had defined {1, 3, 5} above) and the indicator fuction for its members. Klir and Yuan in Fuzzy Sets and Fuzzy Logic p. 6 write
"There are three basic methods by which sets can defined within a given universal set X:
...
3. A set is defined by a function, usually called a characteristic function , that declares which element of X are members of the set and which are not."

Of course one doesn't necessarily have to define crisp sets in this way. But, for any way one defines crisp subsets of a reference set, one *can* rewrite such a definition in terms of a characteristic function.

[So where does it inherent from?]

Unless I've missed it, you've injected 'inherent' into this discussion as you've injected ZFC.

[It still remains the case that sets as we standardly approach them do not come with inherent indicator functions.]

It still remains the case that for all crisp subsets of a reference set I can write or re-write those subsets in terms of an indicator function.

[But the set itself is still the members it obtains]

The empty set qualifies as a set... consequently it qualifies as an object itself. However, the empty set has no members, it has no objects. An object doesn't equal no objects.

[Furthermore, to gain the usage of operations as you used requires us to define more than merely the sets. That is a like mistake you seem to be making with the indicator function. I'm not saying there's anything wrong with it or using it. My point is that it requires more to be spelled out about the investigation we are making.]

For whom, for what, with what purpose, with what standards of proof, with what standards of exactness? This can vary.

[But when I define A={1,2} and B={1,2,3}, we only get relations between A and B because of the members. Our universe of discourse, as denoted, has to contain at least the three names 1,2,3. But let 3=2. Then A=B and (3,0) is wrong. Of course, the indicator function would have to change that fact as the facts have just changed, but my point is that as presented you are imposing a relation of A from B because B becomes the universe and A is in relation to it.]

Sure, but you wouldn't list B as {1, 2, 3} when you wrote it as the domain of discourse or write just A={1, 2}. You'd write B={1, 2} leaving IB(3) unknown, or you'd write A={1, 2, 3}, since the general principle of writing comes as to write things in the least confusing way you can.

[But we do not say A is a subset of B therefore its members are derived from B or something. That wouldn't make sense at all. But that is exactly the kind of thing you seem to be saying by what you said above about separating sets from their members.]

I don't see this.

[If all you have are members and an indicator function, for instance, then we can say we produce sets through the function acting over the members. But then where does the indicator function gets its information about how to place the members?]

I don't follow your question here, or the answer seems obvious enough. We don't need to know how to place the members, since sets have no sequencing. {a, b, ..., z}={z, b, c, w, ... a}. The indicator function has a reference set in its definition.

[If it gets them from the sets, then we had sets all along because sets are the members. There is no transition from members into sets. We just have members and we are drawing imaginary lines around them to enclose them and say "there, that's the set."]

Again, the empty set.

[But there is no behavior involved between sets and members!]

For crisp sets, every set has its memebers as a member. For every crisp set, it comes as determinable that we can say if a potential member of a set belongs to it or not.

[So if the function is inherent in sets and members, then why does it have behavior and sets and members do not?]

Again, where do you get "inherent" from?

[I don't know where you gathered the idea that the characteristic function is inherent in sets.]

Grr... again that word "inherent". Klir and Yuan aren't the only ones to say that we can write all crisp subsets of a set in terms of a characteristic function. Bojadziev and Bojadziev in Fuzzy Sets, Fuzzy Logic, Application p. 104 write "The membership rule that characterizes the elements (members) of a set, A<U can be established by the concept of characteristic function (or membership function ) Ma(x) taking only two values, 1 and 0, indicating whether or not x e U is a member of A:
1 for x e A
Ma(x)= 0 for x ~e A. (5.17)

Hence, Ma(x) e {0, 1}.

Inversely, if a function Ma(x) is defined by (5.17), then it is the charcterstic function for a set A<U in the sense that A consists of the values x e U for which Ma(x) is equal to 1; these values are roots of the equation Ma(x)=1. In other words every set is uniquely determined by its characteristic function."

[There is no probability that just exists in mathematics.]

Also, on this see the Kosko paper "Fuzziness vs. Probability". Since we can talk about probability purely in terms of subsethood and we don't need reality for subsethood, probability can exist just inside mathematics. This doesn't mean we want this or that it should.

Your first example is wrong, since you cannot set up a valid space to measure if it doesn't contain a compliment, for one. The infinite sum or integral across the space must have a density of 1. Therefore, you do not have a probability of the empty set being independent from itself. It doesn't even make sense to talk about it like that. The second example is true. If our space is X then P(X)=1 has to be true. We can have all our density at one point if we want. But that doesn't say anything about probability besides arbitrary denotations. No frequency of something has occurred. No event has happened. There's no chance of likelihood. It's just measure theoretical manipulation of symbols. Theoretical work in the abstraction of probabilities helps us understand the mathematical mechanics--the relationships--that underly what we are trying to model in the application. We determine its success by actual evaluations. How do we know that all these relationships are meaningful? By applying them! You can learn all the rules and theorems you want, but they are meaningless outside of pure mathematical manipulations. As the Wallace and West book point out, the axiomatic system is checked against produced models. A model obtains whenever the axioms provided are all true. True in what? True in our interpretation. Applications are interpretation against real-world information. If we model the mathematical relations as these real-world variables, do we get an isomorphic relation? Then we can gauge the success of our mathematical model. Probabilities are about things in the real world. Frequency is something predicated on phenomena. You don't get phenomena in mathematics. You get logical consequences. You cannot model probabilities or independence, something we talk about in context to real-world causal relations from logical consequence and logical relations. But you say there are mathematical phenomena. What are they? Describe them to me.

As for the empty set, a lack of membership is a property because we can always name a set by what is not included in it (complement). The members are the set and sets are their members. The empty set is zero members and zero members is the empty set. It's like quibbling over whether 0 belongs to the natural numbers or not. To quote from wikipedia's page on natural numbers, "In mathematics, a natural number (also called counting number) can mean either an element of the set {1, 2, 3, ...} (the positive integers) or an element of the set {0, 1, 2, 3, ...} (the non-negative integers). The former is generally used in number theory, while the latter is preferred in mathematical logic, set theory, and computer science." Obviously we're dealing with the latter case. But you say "You can't name members for the empty set. If you do, you don't have the empty set." What is your point? The fact I cannot name the members is what specifies the empty set. The fact I cannot count anything in my wallet right now tells me I have zero dollars in it, or {}εA says A is the empty set. This doesn't undermine anything I've said.

You say fuzzy sets are not well-defined. Is that what some author has told you? Because I am pretty sure from what little I know about fuzzy sets that they are, indeed, well-defined. A set like X={x|x doesn't belong to X} is not well-defined (possibly, depends on how you deal with liar statements). Or Y={x|x is a set} is not well-defined as was shown by Russell's paradox, and led us to move beyond naive set theory to ZF. The set of all sets is not a well-defined set. I'm pretty sure you can properly spell out fuzzy sets. In fact, that seems to be the intuition behind building the characteristic function.

sets equal themselves.

Yeah, and where does the truth of this statement derive? Let us not forget so quickly that truth is under an interpretation in formal systems. Yes, once you have a domain you will have difference in the set, but that is because an interpretation requires a domain. Until then, you're just giving arbitrary denotations. Furthermore, your definition of the set includes symbols, namely "4-6" which are rather meaningless unless we define what negation is, 4 is and 6 is. You might as well have said A={x|x*RHEK}. It would have been just as meaningful. You need to get your head around the fact denotations are separate from connotations or semantics. You assume much when you write these things and that is the point of set theory and mathematical logic or foundations. I suggest picking up a book like Computability and Logic by Boole and Jefferey, or some mathematical logic book that gets into the specifics of these. I just picked up that one so I can train myself for the logic program at Berkeley (they've used it a few semesters, and was recommended by my logic professor, it is too advanced for the stuff taught at my school unfortunately).

Suppose I specify my universe of discourse as R. How in the world does i make sense within my universe of discourse?

I don't follow the question. How does what make sense? That 4 can equal 4+0i? 4+0i is an object that can only be constructed in the complex number system. But 0i equals 0 and can be dropped off. 4+0i = 4. But 4 is ambiguous. If you define a domain, say, like the reals, then you know you are talking about 4 in the reals. But without specifying a domain 4 is not some universal object, it's just a symbol. It has a kind of first-order predicate relation (see the SEP article on Frege), but that says nothing about it in the higher-order structure of sets for which 4 denotes.

Consider our universe of discourse as {0, 1, 2, 3}. What does (2+2) equal? Again, ().

You're still missing a lot. What is addition? The n-ary operation with a domain would give us an algebraic system, like (R,+,*,0,1) is the field of reals. (R,+,e) is the group of reals. You specified a domain, let us call it A. You have (A). That doesn't say anything about 2+2 since it's undefined. I will say it equals 0 and you've got Z_4.

Anyway, as much of what I said is getting rather elementary, I suggest reading more books than just the fuzzy logic ones. From what I saw from the previews, they gloss over some very important stuff in foundations and mathematical logic. They probably assume a certain background of understanding in those areas, for sure. Look at MIT's opencoursware or some university like Berkeley or CMU that have specific programs in foundations and logic and look at the texts they use. Read that stuff to build your basics before getting too deep into advanced stuff like fuzzy logic, otherwise you misinterpret some very basic things you need to have down (like the order of predicate logic used, algebraic systems, etc.)


Back on topic, you seem to be interpreting the transformation the characteristic function provides. You're saying the indicator function is inherent in sets. It isn't. It is something in the transformation (hell, it IS the transformation). Let us say set theory is A and the other theory utilizing characteristic functions is B. Then you have, say, an isomorphism from A onto B. But what is true in A is not true in B. x is true in A and f(x) is true in B, given f is our isomorphism. We can call f(x) b in B. But b is not true in A. f-inverse(b) is true in A. This is what I pointed out in the fact the properties of what is being described in sets is being captured numerically by the characteristic function, but it is no longer dealing with what we have with our standard set theory. The transformation gives us something different, even if they are describing the same thing. If I transform the integers into the natural numbers, I don't say the addition of integers is the same in N (even if denoted the same). They are characteristically different. Any chapter in an algebra book on homomorphisms will cover this. I'm going to the state fair now.

@bryangoodrich - 

[Your first example is wrong, since you cannot set up a valid space to measure if it doesn't contain a compliment, for one.]

I don't see that in the measure theory axioms, but I do see that m("0")=0 where "0" indicates the empty set. Then again I see nothing in the standard axioms of probability theory that says P(x) qualifies as a measure of x. Of course, it's standardily accepted that probabilty qualifies as part of measure theory, but where's the proof? Also, where's the requirement that says we have to a compliment to have a space where we can talk about a probability (or a measure for that matter)?

[The infinite sum or integral across the space must have a density of 1.]

Only if such a sum or integral exists. If we have X as the refernce set and the event, we don't have anything to sum or integrate, even though X has density of 1.

[It doesn't even make sense to talk about it like that.]

I already did and you analyzed it. Did you analyze something nonsensical?

[We can have all our density at one point if we want.]

In which case no sum or integral exists (well maybe some integral exists which can measure a single point).

[No frequency of something has occurred.]

One need not take a frequentist interpretation of probability theory.

[No event has happened.]

No event need happen in reality to compute a probability. The term "event" really just means "subset", since we don't completely talk about things themselves to begin with.

[There's no chance of likelihood.]

I consider chance as a working fiction in probability theory. Kosko's subsethood theorem suggests this.

[It's just measure theoretical manipulation of symbols.]

The mathematical community (from my understanding) writ large, generally thinks probability theory as a branch of measure theory, although they might not study it in such a way.

[How do we know that all these relationships are meaningful? By applying them! ]

Applying to what? One could apply measure theory to develop the subsethood theorem and thus develop probability theory. You seem to mean real-world applications.

[A model obtains whenever the axioms provided are all true.]

The axioms of Newton certainly didn't work out as true. Yet, it still obtained. The axioms just need work out as useful.

[If we model the mathematical relations as these real-world variables, do we get an isomorphic relation?]

Oh dear. There exists NO isomorphism between mathematics and the real-world, the real-world ALWAYS works out as more messy than our mathematical models once we look close enough and/or analyze enough factors. Consider the Earth as oblate spheroid. Does an oblate spheroid work out as a true isomorphism with the shape of the Earth? We have space between the atoms and between the sub-atomic particles of the Earth. Consequently, a continuous oblate spheroid doesn't work, since we have a discrete object. Can we really model the discreteness here? No. This holds for any real-world object. Any sort of model you come up with I can zoom into the atomic level and demonstrate your model as NOT isomorphic with the real-world. Isomorphisms happen between mathematical structures, and only there. The real-world comes as FAR, FAR, FAR more complex than that.

[Probabilities are about things in the real world.]

I understand you have an applied background, but as already indicated not necessarily. Especially if you want to contend that probabilities model randomness or chance in the real-world, since neither chance nor randomness exists in the real-world. Quantum mechanics's use of probability doesn't indicate chance in the real-world due to the subsethood theorem. Do randomness and chance work out as convient working ficitions? Sure. But, do they exist? I maintain the, perhaps philosophical I admit, position of no.

[You cannot model probabilities or independence, something we talk about in context to real-world causal relations from logical consequence and logical relations.]

You can model behaviors in the real-world through logic. One might say that fuzzy rules come as an example par excellence for this.

[But you say there are mathematical phenomena. ]

Did I claim the existence of mathematical phenomena? Did I claim that probabilities exist in the world? No, I didn't.

[As for the empty set, a lack of membership is a property because we can always name a set by what is not included in it (complement).]

Given a reference set we can do this. However, suppose we have a set such as {1, 2, 3}. Again, I'll assume that I don't have a reference set. So, how in the world do I name such a set by what's NOT included in it, since I don't know what the universal set? If I have the reference set {1, 2, 3, 4, 5}=X I can write {x in X: x~=4, x~=5} of C{4, 5}={1, 2, 3}. But, if I have the reference set {1, 2, 3, 4, 5, 6}=X then C{4, 5}={1, 2, 3, 6}. Wikipedia talks about "relative" and "absolute" compliment. You rather clearly mean "absolute compliment". "Absolute complement" gets defined in terms of a reference set. Check it http://en.wikipedia.org/wiki/Complement_(set_theory)

[The empty set is zero members and zero members is the empty set.]

The empty set consists of the set which has no, or zero, members. Suppose we have numbers 2, 3, and 4. Suppose we assume them as free-floating objects or just members. Do we have a set? Well, do we have a collection of members? No, because we didn't specify those free-floating objects to get thought of as a collection. Without them thought of as a collection of objects, we don't have a set... we just have three numbers which we have called "members" (it's hard to use the term "member" without thinking "member of what"?).

[But you say "You can't name members for the empty set. If you do, you don't have the empty set." What is your point?]

You can't define all sets purely by their members, since the empty set has NO members, and thus nothing to define with. Using the property definition indicates a set as something more than just members, but some sort of collection which has elements that satisfy some property.

[The fact I cannot name the members is what specifies the empty set.]

So, you haven't defined the set by its members, but rather by its LACK of members. Consequently, you haven't used members to define the set. You've used a property.

[The fact I cannot count anything in my wallet right now tells me I have zero dollars in it, or {}εA says A is the empty set. This doesn't undermine anything I've said.]

But {} indicates the empty set. Since A is the empty set, I could rewrite your equation as () e (). The empty set IS NOT a member of itself, since if it were the empty set would have a member... namely... the empty set. But again, the empty set has no members, so {}eA where A is the empty set doesn't work. Your statement "{} e A says A is the empty set" gets undermined by the definition of the empty set.

[You say fuzzy sets are not well-defined. Is that what some author has told you? ]

Fuzzy sets have imprecise boundaries (unlike crisp sets). I said "Properly fuzzy sets (those whose membership functions lie in (0, 1)) DO NOT work out as well-defined since you can't tell if an object either belongs or does not belong to the fuzzy set." They don't work out as well-defined in the sense in that you can at one stroke (without some added perspective) tell if we should say f belongs or does not belong to a fuzzy set F. You can "spell out" fuzzy sets in the sense of specifing an element and a memerbship function (which doesn't have to map to the reals necessarily, it could map to the set of fuzzy numbers, or interval numbers for instance). But, some texts use "well-defined" in another sense, and that's the sense I meant. As an example I found this text online which reads:

"The membership criteria for a set must in principle be well-defined, and not vague.
If we have a set and an object, it is possible that we do not know whether this object
belongs to the set or not, because of our lack of information or knowledge. (E.g. “The set of students in this room over the age of 21”: a well-defined set but we may not know who is in it.) But the answer should exist, at any rate in principle."

In prinicple, for properly fuzzy sets (those in (0, 1)) there need not exist an answer to the question "does this object belong or not." We can define such in fuzzy set theory through alpha-cuts for convex fuzzy subsets or as Zadeh did in his original paper by saying x "belongs" if M(x)>0. But, strictly speaking we don't need to do so. http://64.233.167.104/search?q=cache:YXQRZDCToF4J:people.umass.edu/partee/409/
Lecture1%2520Set%2520Theory%2520Sept%25207.pdf
+well-defined+set+theory&hl=en&ct=clnk&cd=4&gl=us&client=firefox-a

Also, concerning members and sets Stoll says this on p. 2 in Set Theory and Logic . "Let us consider Canto'r's concept of the term set and then analyze briefly its constituent parts. According to his definition, a set S is any collection of definite, distiniguishable object of our intuition or of our intellect to be conceived as a whole. The objects are called the elements or members of S.
The essential point of Cantor's concept is that a collection of objects is to be regarded as a single entity (to be conceived as a whole). The transfer of attention from individual objects to collections of individuals objects as entities is commonplace, as evidenced by the presence in our language of such words as "bunch", "covey", "pride", and "flock"."

Also, consider this set {X: X is a subset of itself}. Every set is a subset of itself, so the definition "X: X is a subset of itself" includes every single set, including itself (I can tell you that it belongs to itself). Do you claim this as not well-defined, and how so? I can tell you that every set will belong to it and anything that doesn't belong to it is not a set, so I certainly don't see how it qualifies as not well-defined. Russell's paradox talks about "the set of all sets which are NOT members of themselves."

[sets equal themselves.
Yeah, and where does the truth of this statement derive?]

We assume its truth for crisp sets basically. But, consider a fuzzy set such as
{(3, .5)} and 1-a=c(a). In such a case not{(3, .5)}={(3, .5)}. The complement gives us what is NOT the set. So, the set equals what is NOT the set in such a case. One can then maintain that for fuzzy sets sets need not equal themselves. Of course, though since we have a=~a, and ~a=a, we can still claim a=a or that a set equals itself.

[Let us not forget so quickly that truth is under an interpretation in formal systems.]

Validity doesn't work out as under interpretation.

[Furthermore, your definition of the set includes symbols, namely "4-6" which are rather meaningless unless we define what negation is, 4 is and 6 is. You might as well have said A={x|x*RHEK}. It would have been just as meaningful. You need to get your head around the fact denotations are separate from connotations or semantics.]

You've raised a completely different issue here. Look, the example involved assumes that we know what "4" and "-" and "6" mean already (crisp numbers and regular subraction) and we don't need to define them. Most math courses, exercises, textbooks, teachers, students, etc. do this.

[You assume much when you write these things and that is the point of set theory and mathematical logic or foundations.]

Sure enough, but that works out as irrelevant to the example.

[I don't follow the question. How does what make sense?]

A conceived number i=sqrt(-1) in the domain of discourse R .

[4+0i is an object that can only be constructed in the complex number system.]

So, 4+0i makes no sense in N. Similarly, if we specify our domain of discourse as N for {x: x=4/6}, then x makes no sense. It doesn't exist in N, and consequently {x: x=4/6}={}. It exists in Q or a superset of Q. So, {x: x=4/6}={2/3} (or {x: x=2/3} if you want to talk about equivalence classes... if I remember the logic right). The domain of discourse gives different results for the set {x: x=4/6}.

[But 0i equals 0 and can be dropped off.]

In the complex number system. Well, I guess in the reals also supposing i belongs to the reals. But, see above.

[If you define a domain, say, like the reals, then you know you are talking about 4 in the reals.]

You just defined a universe of discourse. It also gets called a domain of discourse or simply a domain.

[You're still missing a lot. What is addition? The n-ary operation with a domain would give us an algebraic system, like (R,+,*,0,1) is the field of reals. (R,+,e) is the group of reals. You specified a domain, let us call it A. You have (A). That doesn't say anything about 2+2 since it's undefined. I will say it equals 0 and you've got Z_4.]

Fine. All I have to say comes as this: "by '+' I mean ordinary addition." (talking about Z_4 is not ordinary addition and you darn well know it's something different... sure it's addition, but not ordinary addition). So, for the domain {0, 1, 2, 3}, what does {x: x=2+2} represent? The empty set. Why? For ordinary addition, 2+2 doesn't yield a result that exists in our domain.

[Read that stuff to build your basics before getting too deep into advanced stuff like fuzzy logic, otherwise you misinterpret some very basic things you need to have down (like the order of predicate logic used, algebraic systems, etc.)]

I don't accept mathematical concepts as some sort of linearly ordered set. The "Bojadziev and Bojadziev" text referenced requires no more than a knowledge of high-school algebra. They develop fuzzy numbers without even developing fuzzy sets first, but really only from interval numbers (not in a logical sense, but more in a conceptual sense since they have a heavily problem-oriented book). Buckley and Eslami say they require no more than what students usually have after first semester calculus. Klir and Yuan (other than parts on probability theory and possibly some other parts where they mention prerequisite ideas) indicate two references at the beginning, Stoll's book on Set Theory, and the Schaum's book on Set Theory by Seymour Lipschutz and claim Lipschutz's text as less advanced and all that one needs. Of course one has to know things like arithmetic, non-abstract algebra, and some basic things like that, but don't get scared just because a concept gets called "advanced" by some. One need not understand everything in a math text on a first read. As an interesting note, I had never encountered the definition of a metric before I read some books on fuzzy set theory (such as Arnold Kaufmann's). The books reviewed the definiton and I picked something up which works out as useful in other texts. Could I have instead have learned about metrics in a real analysis text (I started browsing some of these after looking at books on fuzzy subset theory)? Sure. But, I didn't need to learn it there.

[You're saying the indicator function is inherent in sets.]

Again, where have I said this? And if I have, haven't the fuzzy subset textbook referenced also done this?

[Let us say set theory is A and the other theory utilizing characteristic functions is B. Then you have, say, an isomorphism from A onto B. But what is true in A is not true in B. x is true in A and f(x) is true in B, given f is our isomorphism. We can call f(x) b in B. But b is not true in A. f-inverse(b) is true in A.]

Maybe that works in general, but I suspect not here. I think it works like this.... The membership relation, or the indicator function work out as isomorphic to truth values (crisp set and classical logic have the same structure). So, if what is true in A is not true in B, then what belongs to A does not belong to B, and consequently we don't have an isomorphism from A onto B, as we previously suppose. In order for us to have an isomorphism from A onto B for membership to characteristic functions, then what is true in A is true also in B, otherwise the isomorphism with classical logic fails. The isomorphism with classical logic doesn't fail, so we the isomorphism from A onto B. False values precede truth values in the sense that F->T, and a member x which doesn't belong precedes a member which does belong in the sense that {x: x does not belong} qualifies as a subset of {x: x does belong} (the empty set precedes any set {a}), and "0" precedes "1". So any comments about assuming an "order" doesn't work, since logic, sets via membership, and sets via characteristic functions all have an order of sorts.

[If I transform the integers into the natural numbers, I don't say the addition of integers is the same in N (even if denoted the same).]

I don't know where I've made an analogous claim exactly. You've used infinite sets which get transformed. If you transform {e, ~e} (belongs, does not belong) into {1, 0} or {T, F} you've transformed finite sets of objects. It's like transforming {-2, -1, 0, 1, 2} into {0, 1, 2, 3, 4}. In such a case -1+0 and 1+2 may yield different results, but given a sequencing of the sets such that we have x1=-2, x2=-1, x3=0, x4=1, x5=2, y1=0, y2=1, y3=2, y4=3, y5=4, we'll have the same index for both results. Both -1+0 and 1+2 yield numbers which have the same position in the sequence. Transofmring (e, ~e) into (1, 0) gives us the same position in the sequence for our values, since we have a similar precedence relationship...
{x: x ~e} subset of {x: x e} and 0<1. With the infinite sets we nothing which precedes everything else or succeeds everythign else. With the empty set we have something which precedes all other sets and with the universe of discourse we have a set which succeds all other sets. 1 succeds 0.

@Spoonwood - 

In which case no sum or integral exists

Um, the sum of x being a single sum of one point which is all the mass. It's discrete so it would be a pmf.

Did you analyze something nonsensical?

Yes. That was the implication. Just because it is nonsense doesn't mean we cannot analyze it, or conclude it is nonsense.

One need not take a frequentist interpretation of probability theory.

No, I just used that interpretation. If you can provide an alternative that discredits what I said then by all means, what would it be?

No event need happen in reality to compute a probability.

So, how do we get probabilities of something without an event occurring? Some Bayesian prior? The SEP article on them adequately shows there is nothing beyond arbitrary claims being made. I brought that out in my blog about probability attacking NQBass7. It has also been shown that frequentist parametric methods model things just the same, if not better, than a Bayesian rationalist approach with priors because they are still going to be evaluated off of some data. We will always require real-world information on which to base our applications in mathematics. Certainly I can say "the probability of X is Y" but that doesn't mean anything about probability in the real world. So unless you mean probabilities exist in some Platonic realm where things have a chance occurrence, then it doesn't make sense.

Did I claim the existence of mathematical phenomena?

Yes, do I need to quote where you did in your last comment? I'm sure you can look it up.

So, how in the world do I name such a set by what's NOT included in it, since I don't know what the universal set?

Naming can be done in numerous ways. Pointing out a case where one method of naming does not work does not discount the whole scheme of denoting a set, so I really don't see your point. In that case no, you cannot define it by what is not in the set, but you also already have it named. That's like you pointing out you cannot name the members of the natural numbers. Okay, that doesn't discount what I said either. You were saying we cannot name the empty set by what I said when it clearly can be since it can still be named. The argument you are trying to make is like saying naming the members is not good enough to name a set since I can claim there is a set you cannot use that method on (e.g., N). But that would be ridiculous to suggest naming A={@,#,$, smiley face} is not naming it.

You seem to mean real-world applications.

Yeah, hence why I was talking about it in terms of applied mathematics. It has to do with real-world applications. It isn't studying or applying to pure mathematics.

The mathematical community (from my understanding) writ large, generally thinks probability theory as a branch of measure theory,

Calculus is properly a branch of measure theory. That doesn't mean differential equations used in the numerous applications has somehow become measure theory or pure mathematics. Furthermore, probability theory can be studied in two different areas. It can be studied in pure mathematics in which case it is much an extension of measure theory, since it is completely abstracted and has to do with the mathematical relations going on. I'm talking about it as an application where probability has actual meaning. I mean, you can use measure theory to define mass over a space, but that doesn't mean mass is properly a mathematical entity. It is capturing something in the real-world. Otherwise, it's just an abstracted symbol, an arbitrary measure. It becomes just symbol manipulation.

The axioms of Newton certainly didn't work out as true.

Newton's axioms don't work out as a proper model, either, unless we talk about ideal conditions. Of course, someone might criticize all of applied mathematics for similar reasons such that no model can ever exist, but this is irrelevant to bring up here.

There exists NO isomorphism between mathematics and the real-world,

Being able to zoom in would suggest you can reduce the planet to nothing but atoms. The relation being modeled on the higher-order object we identify being that collection of atoms still obtains. Therefore, no confusion exists unless you convolute the relation. But modeling works by holding things rather ideal, by holding other variables not required constant so their effects are removed. Looking only at the variables of interest inasmuch as they can be modeled, do they behave as the theory holds. If they do, then we have properly modeled the behavior. If you change the variables, then you've changed the model and it may no longer hold. But that doesn't negate the model that was obtained. This is how much to how scientific models work in general.

since neither chance nor randomness exists in the real-world.

prove it. Of course, you'd probably also want to define those terms.

You can model behaviors in the real-world through logic.

So now you can model the real-world through logic? I would have sworn you just argued against that a moment ago, unless you are redefining what model means. Maybe Wallace and West got it wrong. Nevertheless, I would agree that fuzzy logic does afford more to the field of applied mathematics because it can capture the "messiness" that does exist in modeling reality.

Without them thought of as a collection of objects, we don't have a set... we just have three numbers which we have called "members" (it's hard to use the term "member" without thinking "member of what"?).

Wow, dude. Now you are just reaching. Membership does come with the "of what" clause. You have something that might exist in some first-order predicate calculus, but you are really stretching to say we have this higher-order object (i.e., a member) without the higher-order object it belongs to (i.e., the set). I can certainly have something definite, some individual object defined, say, in some first-order language or theory, and not have a set, at least not in regard to what we might be considering (because the mere fact of predication provides us with a set whether or not some human being is thinking it up at the moment). In fact, you seem to be suggesting we can have magical numbers existing in some Platonic realm that don't belong to a set until human's divine hand of consciousness comes along and conceives of the set those members belong to and then we get a set with members. Are you going to actually suggest that to be the case? Is that how you think set theory works?

You can't define all sets purely by their members, since the empty set has NO members, and thus nothing to define with. Using the property definition indicates a set as something more than just members, but some sort of collection which has elements that satisfy some property.

Awesome, we agree there are multiple ways to define things. I think I even quoted that from the Lighthouse book earlier, but maybe not. In either case, as I have already said, there are numerous ways to denote a set so your argument is rather weak to say "you can't do that in this case." Okay, so what? Listing members is a way to define a set, not the way. Even when we define them by a property or negatively, we still have a set being its members, whether there is zero, one or many. We might even give names to those, like, say, numbers! {} is 0, {{}} is 1, {{{}}} is 2, etc.

So, you haven't defined the set by its members, but rather by its LACK of members. Consequently, you haven't used members to define the set. You've used a property.

And the lack of membership INFORMS us what it is. Having no members informs us the set is empty. It is a valid denotation. If you want to make a counter argument then you need to show me how using any other method from naming its members somehow separate the set from its members. Otherwise, it still holds that our concept of a set ("our" being most mathematicians and how they talk about sets as used throughout mathematics, at least from the past few years I've been at it) is obtained by its membership, i.e., a set is its members. It obtains its property from its membership, i.e., to denote a set by a property is to spell out what members belong to the set due to that property, thus tying the set to its members as denoted through the property that relates them (since a set is a relation over members, whether first or higher-order objects).

But {} indicates the empty set. Since A is the empty set, I could rewrite your equation as () e (). The empty set IS NOT a member of itself,

uh, actually the empty set is a subset of itself. See the wiki page for some of the properties. The empty set is a subset of all sets. Want proof? Take the power function of ANY set and it is in there, even the empty set. The problem with what you are trying to conceive is that the symbols or denotations you are dealing with are some how meaningful. They are not. We only use that to express something. I could have just as well said ___ belongs to A or {} where ___ is nothing. I only used it as a convention to be meaningful to the reader, not as some formal statement. The fact is you cannot say {} belongs to {} since {} is a set and we're talking about members belonging to a set with that notation. If you are saying a set belonging to another set isn't actually correct (though we'd still get it). In that case, you'd say the set is a subset of the other, not that it belongs to it with the epsilon. And yes, {} is a subset of {}. So you are wrong.

You mention A={x|x is a subset of x} is well-defined. How is this well-defined? And Russell talks about A={x| x is a set} (Lighthouse 1965, p. 49). This is the "set of all sets." And yes, every set that is a set will be in there, and every set that is not a set will not be in there, just like the A you provided. That doesn't give us well-defined-ness. People under that superficial (Cantorian) analysis also concluded you could have such a creature. The whole point of Russell's paradox is that it is not well-defined as you can have a set B={x|xεS & x!εx}, for some set S={x|x is a subset of x} and !ε meaning "not belong to." In fact, this is the exact same result in Lighthouse because the set S={x|x is a set} HAS the same outcome of "x is a subset of x" since ALL sets are subsets of themselves. So if you have all the sets and all sets are subsets of themselves, it is the same damn thing as "x is a subset of x" since that ASSUMES x is even a set in the first place. So you are basically just adding another clause about the sets saying "x is a set and a subset of itself" and since the latter part is redundant, it doesn't even need to be there. You just spelled out "the set of all sets" which is proven numerous times over as meaningless. The axiomatization of set theory by, say, ZF or ZFC remove that problem of naive set theory.

You've raised a completely different issue here. Look, the example involved assumes that we know what "4" and "-" and "6" mean already (crisp numbers and regular subraction) and we don't need to define them. Most math courses, exercises, textbooks, teachers, students, etc. do this.

Look, you obviously don't understand mathematical foundations then if you want to go that route. You CANNOT use operations in a definition without first spelling out an algebraic system for the damn thing to exist in. It makes NO SENSE to talk about "regular addition" as if that is some universal thing that all mathematicians can use because we always do it and it is in ever math course, textbook, etc. Pick up any introductory book on mathematical logic, or look at the google book previews for some of the books I've suggested and educate yourself on that fact. When you name a set it is meaningless. To say S={x|x=8+3} without spelling out that we're working in Z, thus obtaining its group properties at least (AN ALGEBRAIC SYSTEM!) then that statement is meaningless unless we assume such a thing to already be true. If we're working in Z_2, then you really have S={x|x=1}. So what? Either algebra is worthless to involve here, I'm on a whole other issue, or you think we can just assume wtf-ever we want. In either case, you are wrong again. If you want to say 1 doesn't equal 8+3 so I cannot do that substitution, then why not? 8+3 NAMES a single member, by our most general assumptions (that addition being used by that symbol is binary). What does it name? that depends on the damn algebraic system which needs to be specified. It means 11 in Z, it means 1 in Z_2, it means 2 in Z_3, it means 3 in Z_4, etc. One of those is an integral domain, two of them fields and one just a boring ring with zero divisors. They are all different and it does matter which algebra is being used. Inadvertently if we assume we are in Z, then you've already spelled out a domain (i.e., universe of discourse) that S belongs to.

I would read the rest of this, but it is late and I'm tired of explaining basic stuff you can pick up on your own. Read an algebra book or any of the suggested texts or text subjects I've offered, or find one for yourself from the syllabus of a good university course on the matter. It is rather trying to discuss mathematics, especially advanced stuff or unique and specific issues when you don't grasp the very basics used in mathematics. This is why advanced courses require the background of having already been exposed to algebra and analysis and set theory and formal mathematics and logic (though, the latter two or three may often be summed up in a single overview course, such as at my college which doesn't deal in foundations at all. Unlike Berkeley which incorporates it in their algebra courses and have specific courses on the advanced topics, both at the undergrad and graduate level).

But in closing, you ask

Again, where have I said [the indicator function is inherent in sets]?

Well, let us see:

Using the universe and using the membershp relation come as inherent in sets ... Membership comes as inherent in sets. Since membership determines the behavior of the indicator function, this means we can write every set in terms of its membership function, since you can write every set in terms of its members.

Are you going to say you weren't spelling out that the universe and membership as inherent in sets, and that this then informs our indicator function so every set inherits the value of its membership as specified by the indicator function? Seems to be what you just said and were trying to do. My point was that we have a transformation which is not the same thing as what we have in the former, even if the indicator function maps that perfectly (isomorphism). If you want to go back on that, then we can happily agree that an indicator function is not inherent in set theory. It is built upon it to describe the property of membership in a numeric way after mapping it, in the one case, to Z_2. Thus, we can describe any subset of a set with an indicator function that maps its membership in terms of a collection of Z_2 elements.

I will also add that much of undergrad math can be learned without the proper prerequisites, but one still lacks understanding necessary concepts. As you have done, you filled in the gaps by your own interpretation of specific terms. There are many things that might be said that have a specific meaning that you take loosely when you might not. Nevertheless, consider the fact you take "ordinary addition" to be meaningful. It isn't. What is ordinary about it? Oh, that's right, it being a binary operation ON THE INTEGERS. Look up the wiki (or other source) on logical signatures. Whether it's just the required domain and statement key in first-order predicate logic, or a relation and a domain for an algebraic system (along with special non-logical constants), we still require that it all be spelled out in some manner. You are utilizing an algebraic system and you DON'T UNDERSTAND IT. That is my point. You think you can talk about addition without specifying the god damn algebraic system that we must be in to use that. Once you have addition and you have it operating on the Z domain, then you also have the identity, and you have an algebraic system (R,+,0). That doesn't just manifest whenever we spell it out and only then. It is there once you are utilizing those elements in your investigation. If you say S={x|x=2+2} but S also equals {1,2,3}, then you cannot say you are working only in Z because you'd be stating a contradiction. It isn't the mathematics that is wrong, it is the human (your's) error. If + is "ordinary" which usually implies working on the integers, then S cannot meet both of those denotations, period.

@bryangoodrich - 

[Nevertheless, consider the fact you take "ordinary addition" to be meaningful. It isn't. What is ordinary about it? Oh, that's right, it being a binary operation ON THE INTEGERS.]

No, when you talk about "ordinary addition" you mean the sort of addition that accountants, high school students, and cashiers do. Fractions get added well before high school for most students.

[You think you can talk about addition without specifying the god damn algebraic system that we must be in to use that.]

High school, middle school, and elementary teachers talk about 'ordinary addition' without specifying the algebraic system that they use. So, rather clear you CAN talk about 'ordinary addition' without specifying an algebraic system. Newton used 'ordinary addition' at a time when neither algebraic systems nor set theory existed. So did THOUSANDS of other mathematicians. The Pythagoreans developed arithmetical theorems long, long before set theory or algebra. Some of them concerned addition. You CAN view addition in the light of set theory and/or algebraic systems, but you certainly do NOT need to do so. Kronencker also comes to my mind as a promeninent here, since he totally rejected Cantor's set theory but said things like "God made the integers; all else is the work of man." Ramanajuan made PLENTY of arithmetical statemens without using algebraic systems or set theory. Again, you CAN AND MAY view arithmetic this way, but you certainly do NOT need to do so.

[Once you have addition and you have it operating on the Z domain, then you also have the identity, and you have an algebraic system (R,+,0).]

You don't need an identity to talk about addition on Z+. You can still add 3 and 6 and get a result in Z+.

[If you say S={x|x=2+2} but S also equals {1,2,3}, then you cannot say you are working only in Z because you'd be stating a contradiction.]

Sure, but you can always work on a subset of Z, such as {1, 2, 3}. In such a case the notion of addition in the ordinary sense of the term still makes sense since you can say that 1+2=3 and 1+1=2, so addition does have some instances where it works on the reference set. For instance, say I have either two one dollar bills or a two dollar bill and a one dollar bill in an urn. How much money lies in the urn? We only need the subset of Z {1, 2, 3} as our reference set to add the possible combinations of bills to together to say that we have either 2 dollars or 3 dollars in the urn. We DO not need the entire Z as our reference set, because of the situation of our problem.

[It isn't the mathematics that is wrong, it is the human (your's) error.]

Haha... you've just said that ever mathematician in existence who did arithmetic before the 20th century qualifies as wrong did it wrong. Talk about arrogant.

[If + is "ordinary" which usually implies working on the integers, then S cannot meet both of those denotations, period.]

No, it doesn't. In the ordinary everyday world I can add fractions of pies together, such as 1/3 of a pizza and 1/2 of a pizza and get 5/6 of a pizza.

[In which case no sum or integral exists

Um, the sum of x being a single sum of one point which is all the mass. It's discrete so it would be a pmf. ]

I forget that the integral from a to a of a function dx equals 0 here.

[No, I just used that interpretation. If you can provide an alternative that discredits what I said then by all means, what would it be?]

It won't discredit it exactly, but it shows it less fundamental. One can derive the conditional probability equation from considerations given in Kosko's geometric proof of the subsethood theorem. I take that as meaning that the subesthood interpretation comes as more elementary than any other interpretation really. The subesthood interpreation says that probability describes the degree of the whole in the part. Kosko's intro. to his paper says "A new geometric proof of the Subesthood Theorem is given, a corollary of which is that the apparently probabilistic relative frequency na/N turns out to be the determinstic subsethood S(X, A), the degree to which the sample space X is contained in its subset A. So the frequency of successful trials is views as the degree to which all trials are successful." None of this means one should necessarily learn probability theory this way though, since using measure theory like this (as Kosko does) makes things more abstract.

[So, how do we get probabilities of something without an event occurring?]

You compute it through measure-theoretic considerations.

[The SEP article on them adequately shows there is nothing beyond arbitrary claims being made.]

SEP article?

[We will always require real-world information on which to base our applications in mathematics.]

For applications to reality, sure.

[Certainly I can say "the probability of X is Y" but that doesn't mean anything about probability in the real world. So unless you mean probabilities exist in some Platonic realm where things have a chance occurrence, then it doesn't make sense.]

You don't need "chance" for probability theory. Probabilities do NOT actually talk about chance. You won't understand what I say until you realize and interpret what I say in the framework that when I talk about a probability I don't mean what you mean by a "probability" in terms of a measurement of a chance occurance.

[Yes, do I need to quote where you did in your last comment? I'm sure you can look it up.]

I would have preferred that. Searching the text, I did use such a phrase. By that term I merely mean mathematical objects of our consciousness. I guess it's a bad phrase, since I don't mean mathematical physical objects as one might think.

[Naming can be done in numerous ways. Pointing out a case where one method of naming does not work does not discount the whole scheme of denoting a set, so I really don't see your point.]

It DOES discount the proposition that we can talk about ALL sets in terms of a naming scheme. You originally said "The fact that I can name (i.e, denote) the set by its members..." In other words, for any given set S, you can name it by its members. You simply can't do that for any given set S.

[That's like you pointing out you cannot name the members of the natural numbers.]

So, for any infinite set you can't name the set by naming its members, since you can't name all the members of the natural numbers. So, your method doesn't work for plenty of sets.

[You were saying we cannot name the empty set by what I said when it clearly can be since it can still be named.]

You can name the set, but you can't name the set by (naming) its members.

[But that would be ridiculous to suggest naming A={@,#,$, smiley face} is not naming it.]

I didn't say you couldn't ever use the list method (naming a set by its members). I just said you can't always use. NOT ALWAYS DOES NOT MEAN NEVER. Sure enough, it does in the framework of two-valued logic. But, in plain English it doesn't. It doesn't work out that way in most other languages either. I think I wrote in plain English there.

[That doesn't mean differential equations used in the numerous applications has somehow become measure theory or pure mathematics.]

The equations themselves, yes. What those equations refer to, how to develop them, how to use them to analyze physical phenomena, etc. no. Like probability theory, one can study differential equations in the realm of pure mathematics.

[I'm talking about it as an application where probability has actual meaning.]

If a pure mathematician studies probability theory in the context of measure theory and it has personal meaning to him, then it has actual meaning to the mathematician, since the pure mathematician counts as a person.

[It is capturing something in the real-world.]

No, it models something in the real-world. It does not tell us about things-in-themselves.

[Newton's axioms don't work out as a proper model, either, unless we talk about ideal conditions.]

Every single model works this way. There exists no one-to-one onto correspondence between the real-world and our modeling of the phenomena. If you really want to make this requirement, nor proper models exist... none at all. So, why criticize Newton's model on such grounds? (the question indicates the reasoning here relevant).

[Being able to zoom in would suggest you can reduce the planet to nothing but atoms.]

No, because you have to also account for the relations between atoms.

[The relation being modeled on the higher-order object we identify being that collection of atoms still obtains.]

The world DOES not have such a "higher-order" object. The Earth exists at each the order of the entire Earth, the order of continents, the order of atoms and so on.

[prove it. Of course, you'd probably also want to define those terms.]

I don't think you can disprove the existence of chance or randomness. I don't think you can prove the existence of such either.

[So now you can model the real-world through logic?]

I didn't say you could model everything that way.

[Nevertheless, I would agree that fuzzy logic does afford more to the field of applied mathematics because it can capture the "messiness" that does exist in modeling reality.]

Only to a certain degree less than unity.

[Membership does come with the "of what" clause.]

Then we have a reference set for membership. In other words, we can't just have members by themselves.

[In either case, as I have already said, there are numerous ways to denote a set so your argument is rather weak to say "you can't do that in this case." Okay, so what? Listing members is a way to define a set, not the way.]

Then the claim "the set is its members" comes out false, since we have a set which we can't specify purely in terms of its members.

[Even when we define them by a property or negatively, we still have a set being its members, whether there is zero, one or many. We might even give names to those, like, say, numbers! {} is 0, {{}} is 1, {{{}}} is 2,]

Fine, let's give the number 0 to {}. The set has 0 members. If a set "is its members" and consequently we have to specify any given set by its members, we have NOTHING to specify the set {} with. We can't define it. So, the "set is its members" notion fails for the empty set. Consequently, there exists a set such that it is NOT its members.

[uh, actually the empty set is a subset of itself.]

Yes, but {} e {} doesn't say the empty set is a subset of itself, {} < {} says that. {} e {} says the empty set is a member of itself. The membership and subset relationship differ.

[I only used it as a convention to be meaningful to the reader, not as some formal statement.]

If you mean 'e' as subsethood, you didn't use the convention, since by convention 'e' means "belonging to" not subsethood.

[And yes, {} is a subset of {}. So you are wrong.]

And you've accused me of conflating subset and membership? Look, here's the whole passage:
"But {} indicates the empty set. Since A is the empty set, I could rewrite your equation as () e (). The empty set IS NOT a member of itself,

uh, actually the empty set is a subset of itself. See the wiki page for some of the properties. The empty set is a subset of all sets. Want proof? Take the power function of ANY set and it is in there, even the empty set. The problem with what you are trying to conceive is that the symbols or denotations you are dealing with are some how meaningful. They are not. We only use that to express something. I could have just as well said ___ belongs to A or {} where ___ is nothing. I only used it as a convention to be meaningful to the reader, not as some formal statement. The fact is you cannot say {} belongs to {} since {} is a set and we're talking about members belonging to a set with that notation. If you are saying a set belonging to another set isn't actually correct (though we'd still get it). In that case, you'd say the set is a subset of the other, not that it belongs to it with the epsilon. And yes, {} is a subset of {}. So you are wrong."

Here's my last statement (as quoted) and your last statement

"The empty set IS NOT a member of itself" -Spoonwood

"And yes, {} is a subset of {}. So you are wrong." -BryanGoodRich

I did NOT say the empty set is not a subset of itself, I said that it was NOT a member of itself. Memership does NOT equal subsethood.

[The whole point of Russell's paradox is that it is not well-defined as you can have a set B={x|xεS & x!εx}, for some set S={x|x is a subset of x} and !ε meaning "not belong to."]

This talks about membership, NOT subsethood. My statement talks about subsethood "You mention A={x|x is a subset of x} is well-defined." Again, membership and subsethood work out as distinct.

[In fact, this is the exact same result in Lighthouse because the set S={x|x is a set} HAS the same outcome of "x is a subset of x" since ALL sets are subsets of themselves. So if you have all the sets and all sets are subsets of themselves, it is the same damn thing as "x is a subset of x" since that ASSUMES x is even a set in the first place.]

I'll grant you that, but that is NOT Russell's paradox which talks about MEMBERSHIP AND NOT SUBSETHOOD. Also, in this case x DOES belong to x unlike in Russell's paradox where x does NOT belong to x. Maybe that doesn't work, but consider this also:

"There are set theories known to be consistent (if the usual set theory is consistent) in which the universal set V does exist (and V \in V is true). In these theories, Zermelo's axiom of separation does not hold in general, and the axiom of comprehension of naive set theory is restricted in a different way. Examples of such theories are the various versions of New Foundations which are known to be consistent and systems of positive set theory." http://en.wikipedia.org/wiki/Set_of_all_sets

[You CANNOT use operations in a definition without first spelling out an algebraic system for the damn thing to exist in.]

Yes, you can. Euclid, Newton, etc. did so. You may not do so in some context, but nothing makes such impossible.

[Pick up any introductory book on mathematical logic, or look at the google book previews for some of the books I've suggested and educate yourself on that fact.]

This consists of no more than a questionable appeal to authority... that of standard textbooks. Why questionable? They often say things like that the prinicple of contradiction must hold. Take Copi's book "When understood in the sense in which it is intended, the principle of contradiction is unobjectinable and perfectly true." p. 390 of Introduction to Logic with the principle of contradiction as "The principle of contradiction asserts that no statement can be both true and false." Well, statements do NOT have to have truth values in {0, 1} they merely have to have a truth value of some sort, so statements can be both true and false. Try and construct your own arguments, please.

[To say S={x|x=8+3} without spelling out that we're working in Z, thus obtaining its group properties at least (AN ALGEBRAIC SYSTEM!) then that statement is meaningless unless we assume such a thing to already be true.]

So, then you have to have a reference set or universe of discourse or domain of discourse or more tersely even a domain... here Z. If you have that, then you can write an indicator function for any set you'll talk about, since you can only logically talk about subsets of the reference set, otherwise you've changed the subject.

[If we're working in Z_2, then you really have S={x|x=1}.]

So, you have a different reference set.

[that depends on the damn algebraic system which needs to be specified.]

In other words the reference set, since for each algebraic system we have a different reference set of possible answers. For Z we have {1, ..., 11}, for Z_2 we have {1, 2}. For Z_3 we have {1, 2, 3}, for Z_4 we have {1, 2, 3, 4}.

[Inadvertently if we assume we are in Z, then you've already spelled out a domain (i.e., universe of discourse) that S belongs to.]

So it makes no sense to talk about sets free of a universe of discourse. Since we always have a universe of discourse when *can* always specify an indicator function for a given subset of that universe of discourse. Since the universe of discourse qualifies as a subset, we can speficy an indicator function for the universe of discourse. So, we can specify an indicator function for all subsets of any given universe of discourse... in other words for every single set.

[Are you going to say you weren't spelling out that the universe and membership as inherent in sets and that this then informs our indicator function so every set inherits the value of its membership as specified by the indicator function?]

The universe and membership inform our indicator function. One can write crisp sets without the indicator function, such as {1, 2, 3} with {1, 2, 3} as its refernce set. But, one can also write them with it. The correspondence between membership and the indicator function does hold for all crisp sets. If Ia(x)=1, then membership exists. If Ia(x)=0, then memebership does not exist. If membership exists, then Ia(x)=1. If membership does not exist, then Ia(x)=0. One implies the other, because Ia(x) by definition belongs to {0, 1}. Hopefully, the wikipedia comes out correctly, "The indicator function of a subset A of a set X is a function
\mathbf{1}_A : X \to \lbrace 0,1 \rbrace \, " http://en.wikipedia.org/wiki/Indicator_function. If x belongs to a, then Ia(x)=1. We suppose x belongs to a, and consequently Ia(x) also holds as true. So, it holds as true that either x does not belong to a or Ia(x)=1. Consequently, we have Ia(x)=1 or x does not belong to a as true. In which case we have that when it is not the case that Ia(x)=1 implies x does not belong to a, also as true. Now, do NOT use contraposition since that won't yield anything in another form than what we origianlly said. It holds as not the case that Ia(x)=1 when Ia(x)=0 since Ia(x)->{0, 1}. In other words, if Ia(x)~=1, then Ia(x)=0. So, we rewrite "when it is not the case that Ia(x)=1 implies x does not belong to a" as Ia(x)=0 implies x does not belong to a.

Similarly, suppose we have x not belonging to a implies Ia(x)=0 as true. In other words either it is not the case x does not belongs to a or Ia(x)=0 works out as true. If it is not the case that x does not belong to a, then it is the case that x belongs to a. So, either x belongs to a or Ia(x)=0 works out as true. So, Ia(x)=0 works out as true or x belongs to a. So, Ia(x) not equal to 0 implies x belongs to a. If Ia(x) does not equal 0, then since Ia(x)->{0, 1}, Ia(x)=1. This means we rewrite the above as Ia(x)=1 implies x belongs to a.

See any flaws here?

A particular behavior of the indicator function implies the particular behavior of the membership relation (or lack thereof if you like) as demonstrated above. So, it follows that one *can* describe crisp sets in terms of collections of indicator function, since one can describe sets in terms of collections of members, since the particular behavior of Ia(x) DOES tell you the particular behavior of the membership relation given a reference set X. Every (crisp) set can inherit its membership relation from the beahvior of the indicator function if you choose to write things that way. This holding for fuzzy sets comes as another matter, since the notion of membership doesn't have such a cut and dry definition.

[It is built upon it to describe the property of membership in a numeric way after mapping it, in the one case, to Z_2.]

I don't have to write what I mean by the indicator function in terms of numbers. Say we have Ia(x)->{y, z} with y<z and with the stipulation that ~e<e (not belonging precedes belonging) no more. Here come the above demonstrations rewritten:

"If x belongs to a, then Ia(x)=z. We suppose x belongs to a, and consequently Ia(x) also holds as true. So, it holds as true that either x does not belong to a or Ia(x)=z. Consequently, we have Ia(x)=z or x does not belong to a as true. In which case we have that when it is not the case that Ia(x)=z implies x does not belong to a, also as true. Now, do NOT use contraposition since that won't yield anything in another form than what we origianlly said. It holds as not the case that Ia(x)=z when Ia(x)=y since Ia(x)->{y, z}. In other words, if Ia(x)~=z, then Ia(x)=y. So, we rewrite "when it is not the case that Ia(x)=z implies x does not belong to a" as Ia(x)=y implies x does not belong to a.

Similarly, suppose we have x not belonging to a implies Ia(x)=y as true. In other words either it is not the case x does not belongs to a or Ia(x)=y works out as true. If it is not the case that x does not belong to a, then it is the case that x belongs to a. So, either x belongs to a or Ia(x)=y works out as true. So, Ia(x)=y works out as true or x belongs to a. So, Ia(x) not equal to t implies x belongs to a. If Ia(x) does not equal y, then since Ia(x)->{y, z}, Ia(x)=z. This means we rewrite the above as Ia(x)=z implies x belongs to a."

You don't need Z_2 which refers to numbers. You just need spell two unequal marks "a, b" with an order relation a<b and a supposition that ~e<e.

[Thus, we can describe any subset of a set with an indicator function that maps its membership in terms of a collection of Z_2 elements.]

You can do that sure, but the idea at work comes out more general.

@Spoonwood - 

If you do addition on a set {1,2,3} then you do not have a binary operator. Part of the definition is that it is closed. But then you'd have to somehow define one of those variables as the identity. Otherwise, what would 3+2 equal? Yes, you can talk about addition but it is not an operator on a set. You are very loosely just throwing it into the property of a set as if it is meaningful, as if 3+4 means something without specifying that it is an operator on the integers. If you don't realize that all the basic stuff done in earlier mathematics classes takes much of this for granted, then that might be why you seem to continue to overlook it.

SEP is the stanford encyclopedia of philosophy. I've referenced it before with links and it (was) on my Blog. You can now find it on my website http://www.bryangoodrich.com/links.html I think.

[If you do addition on a set {1,2,3} then you do not have a binary operator. Part of the definition is that it is closed.]

O.K., but you can still do ordinary addition, since ordinary addition doesn't assume closure.

[But then you'd have to somehow define one of those variables as the identity. ]

You don't need to have an identity to say 1+1=2.

[Otherwise, what would 3+2 equal?]

If we have {1, 2, 3} as our reference set, it doesn't exist.

[Yes, you can talk about addition but it is not an operator on a set. You are very loosely just throwing it into the property of a set as if it is meaningful, as if 3+4 means something without specifying that it is an operator on the integers.]

You don't need the whole set of integers to add 3 and 4, you only really need {3, 4, 7}.

[If you don't realize that all the basic stuff done in earlier mathematics classes takes much of this for granted, then that might be why you seem to continue to overlook it.]

It doesn't get taken for granted necessarily. People did addition before set theory existed, before the concept of closure existed, before the concept of an identity existed for addition, since 0 didn't exist until sometime after the birth of Jesus. They simply couldn't take set theory for granted, or that the set of counting numbers worked out as infinite, or closure, because these concepts didn't exist at the time.

[The SEP article on them adequately shows there is nothing beyond arbitrary claims being made.]

This one? http://plato.stanford.edu/entries/probability-interpret/

It reads
"‘Probability’ is apparently, among other things, a modal concept, plausibly outrunning that which actually occurs, let alone that which is actually observed."
The subsethood theorem suggests otherwise.
It reads
"Applicability. The force of this criterion is best expressed in Bishop Butler's famous aphorism, “Probability is the very guide of life.”"
No. Probability does not come as the only characterization of uncertainity. It does NOT deal with all types of uncertainty.
It says
"Applicability to rational belief: an interpretation should clarify the role that probabilities play in constraining the degrees of belief, or credences, of rational agents. Among other things, knowing that one event is more probable than another, a rational agent will be more confident about the occurrence of the former event."
No. Look, suppose some person A starts getting worried about their stash of 100,000 dollars getting stolen. They have it all under their mattress. They simply won't put it in any bank, or suppose there exist no reliable banks this person can access (this happened during The Great Depression in some areas I'd think). "A" doesn't usually have visitors. They have a heavy duty lock on their room and only one copy of the key for the lock which they carry with them at all times. Only person A goes into the room. Without this lock, the probability of their house NOT getting broken into equals the probability of their money getting stolen and let's say it equals .99. The probability of the house getting broken into equals .01. However, even when confronted with relevant data that indicates this probability, the person A still feels more confident that his/her house will get broken into at some point than not. The person acts in accordence with this confidence. Does, this person no longer act like a rational agent (the term "agent" has the 'nt' ending which behaves like 'ing'... meaning 'agent' indicates a 'doing') if A still wants the lock on his/her room there and to carry the key on him/her physically at all times, since A still believes the probability of his/her money getting stolen than not? No. The person acts rationally (even though she/he has an incorrect belief about a relative frequency of like people), because the degree of harm simply comes as too high. So high that really irregardless of the probability involved the risk comes as all too high that there exists no way that person should give up carrying the key and having that lock. Don't think this some isolated example, because...

Well, what's the probability that someone would get hit in the head in a batting cage by a baseball? You could probably do all sorts of calculations from how machines like this work, or collect sample data, or whatever, and rather clearly show that in very, very few cases this will actually happen. So, we have a very low probability of getting hit in the head in a batting cage. But, the risk comes as so high, that if one doesn't wear a helmet, one doesn't act rationally. One can believe the probability of getting hit in the head high, and still act rationally if you wear your helmet. Probability does NOT, and SHOULD NOT guide us here. The probability of a biker falling off his bike and cracking his head open also comes as low. But, the risk comes out as high. The probability of someone getting in a car accident during a 10 minute drive comes as low, since most of the time when we drive accidents don't happen. But, the risk comes as so high that in many states you legally have to have car insurance.

So, taking the SEP article as definitive certainly doesn't work. The SEP article also doesn't discuss the subesthood interpretation of probability. Maybe similar ideas get covered in there somewhere, but still that article has some major difficulties.

I don't know about the value of that article in general, but since it says something like "A fair coin that is tossed a million times is very unlikely to land heads exactly half the time; one that is tossed a million and one times is even less likely to do so!" it totally mis-represents the frequentist interpretation, since a million does not like close to infinity in the sense that lim n->oo. In fact, no finite numbers lie close to infinity in that sense, since for any finite n in N, n>10, I can just raise it to its own power, i.e. take n^n, and get a finite number on the next order larger than n. And again, what about the subsethood interpretation?

@Spoonwood - 

You compute it through measure-theoretic considerations.

Compute WHAT? Some arbitrary value? That is no different than just showing the abstract theorems with symbols. There's nothing meaningful there.

O.K., but you can still do ordinary addition, since ordinary addition doesn't assume closure.

You still haven't defined what "ordinary addition" even is. So your example that you can name something in such a way that it goes outside of the set, e.g., R={1,2,3} and R' is a subset of R such that R'={x|x=2+2} is supposed to show what? That you can arbitrarily put together symbols with supposed meaning like "ordinary addition" and have a problem like lack of closure? I don't see what is supposed to be expressed let alone anything meaningful.

You don't need to have an identity to say 1+1=2.

You apparently don't even realize the assumptions that go into that statement. Maybe you'd notice them if you could define addition as it is being used, but then you'd have to spell at that out and realize how asinine a statement like "ordinary addition" is.

If we have {1, 2, 3} as our reference set, it doesn't exist.

It doesn't? How to you determine that? Because you've DEFINED ADDITION THAT WAY? The set of nonzero elements in Z_4 is that domain. Addition would say that 2+3=1. 1 is still in the set. Wow, it exists. Why? Because it depends on the system that the operation is operating on. Apparently you think an operator can exist in that free floating place with the free floating members. Are you a Platonist?

You don't need the whole set of integers to add 3 and 4, you only really need {3, 4, 7}.

No, but you need to spell out what addition is. You need to realize that when you say 3+4 it IS 7. It isn't that they are different values, that + symbol SAYS that 3+4 is an element, just like I can say a function mapping from A to B where a belongs to A annd b belongs to B that a is in A and f(a) is in B. f(a) is not separate from the b that it happens to equal. They are names. 3+4 denotes 7 iff the binary operator acting on the two elements equals 7. But that requires it to actually be an operator and maps over a domain. Your use of addition here in this contrived example is basically saying "addition is meaningless beyond the fact I'm going to name 7 as 3+4." Addition as you are using it really has no value as an operator. It's just an arbitrary symbol. You cannot take it and apply it to, say 3+7 since that makes no sense as you used it. What does 3+7 name? 10? It's not in your domain. Addition isn't even an operator anymore. If it is, and you are using it standardly as over most groups, then you have to include the whole domain of, say, the integers. Otherwise, it's like trying to talk about Peano arithmetic and saying the successor function only acts on the variables i named in the domain I named. You don't seem to understand that the operator helps to generate the domain. A set with an operator produces an algebraic system. Therefore, (Z,+,0) tells us what the members of Z are because for any member in Z we can use + to find new members in Z because addition is closed. If you take that away, then you have to accept it as either arbitrarily meaningless or not an operator. If your domain is somehow limited to those three elements and it happens to work as you want to define it on one element such that you can name 3+4 as 7, it doesn't mean you can even talk about 7+3 since that doesn't name anything. It doesn't just breach some wall and fall into some other magical free floating set or something.

since 0 didn't exist until sometime after the birth of Jesus.

Last time I checked the Hindus had the concept that 0 derived from well before Jesus.

what about the subsethood interpretation?

What about it?

[Compute WHAT? Some arbitrary value? That is no different than just showing the abstract theorems with symbols. There's nothing meaningful there.]

Not if a mathematician finds such meaningful.

[You still haven't defined what "ordinary addition" even is.]

No I haven't, but it's well understood, so that doesn't make for a relevant criticism.

[So your example that you can name something in such a way that it goes outside of the set, e.g., R={1,2,3} and R' is a subset of R such that R'={x|x=2+2} is supposed to show what?]

It shows that x=2+2 does not exist in R.

[That you can arbitrarily put together symbols with supposed meaning like "ordinary addition" and have a problem like lack of closure?]

I don't accept your feigning that you lack an intuitive understanding of what "ordinary addition" means.

[You don't need to have an identity to say 1+1=2. You apparently don't even realize the assumptions that go into that statement.]

Such as what? The statement existed well before set theory, algebra, the notion of an identity. Maybe I got the notion of 0 wrong. But, you can find statements in Babylonian texts for example which predate the creation of 0, which say things like 1+1=2. Do you mean to say that they assumed set theory, an identity, an algebra of sorts? All of that comes as absurd, since NONE of those concepts existed for the ancient Babylonians, or the ancient Egyptians.

[Maybe you'd notice them if you could define addition as it is being used, but then you'd have to spell at that out and realize how asinine a statement like "ordinary addition" is.]

In order to define it as it's being used I would have to define it in such a way that indicates how an ordinary brain works when it sums together two objects. Set theory does NOT capture this, since a binary process does NOT take place when you add 1+1 or 156+567 in the ordinary sense... it's far more complicated than that. I don't think you can define addition in terms of how the brain uses it. However, we still understand enough to know something about how do such addition, to explain it to children using examples like counting sticks or rocks or counting parts of pizza pies. You may fail to understand what I mean formally here, but I don't need to explain such formally, since you already understand ordinary addition at a much, much more intuitive level.

[It doesn't? How to you determine that? Because you've DEFINED ADDITION THAT WAY? The set of nonzero elements in Z_4 is that domain. Addition would say that 2+3=1. 1 is still in the set.]

Z_3 does NOT consist of ordinary addition, becuase in ordinary addition no "3" for "Z" exists. This doesn't mean that ordinary addition operates on the countably infinite set Z, because in ordinary life we don't work with infinite sets of pants, dollars, socks, trees, or whatever else we want to add together. I also didn't define '+' as operating on Z_4. I defined it as ordinary addition, which means that it when it operates on the natural numbers, it works analogously to counting. If we can only count up to three, as the domain {1, 2, 3} specifies, then 2+2 doesn't make any sense, since we have to break the analogy with counting to get anything but 4. If you know how to count, you know how to do ordinary addition on a subset S={1, ..., n} of N. Do you want to say you don't know how to count?

[Why? Because it depends on the system that the operation is operating on.]

That's only if you define '+' as a binary opeator +:SxS->S where S indicates the reference set. The notion of addition and ordinary addition, existed LONG, LONG before that of closure and binary operations, as I've already said. If a+b just indicates some process that gives a result the same as counting b times from a, without closure or a binary operation, which tells you what ordinary addition basically means for natural numbers, then + doesn't depend on the system that you operate on in such a way that 2+2=1 for the universe of discourse {1, 2, 3}. In such a case, 2+2 doesn't work as having any solution.

[Apparently you think an operator can exist in that free floating place with the free floating members.]

You can assume such and see what consequences follow from such an assumption.

[You need to realize that when you say 3+4 it IS 7. ]

No. 3+4 indicates two objects with a function operating on them (or something adding them if we don't have the notion of function yet). 7 indicates one object. Two objects are not one object. These sort of statements indicates why even in mathematics I try to avoid "be" verbs. 3+4 EQUALS 7. 1 piece of pizza in proportion to 3 pieces of pizzas EQUALS 3 pieces of pizza in proportion to 9 piecese of pizza, but 1 piece of pizza in proportion to 3 pieces of pizzas IS NOT THE SAME AS 3 pieces of pizza in proportion to 9 pieces of pizza. 1/3=3/9 sure, but 1/3 IS NOT 3/9. You'll have a very hard time convincing me otherwise.

[f(a) is not separate from the b that it happens to equal.]

f(a) indicates the process of a function operating on an object a. b indicates an object. A process operating on an object does not equal an object, unless something which changes can equal an object. You can say that sure, but in such a case an object no long becomes something constant, but rather something changing, because the notion of process indicates change. This would mean that f(a)=4 indicates 4 as some sort of (changing) process. But, we assume 4 as some sort of constant. So it doesn't change. Thus, it doesn't work out as a process. But, we indicated it a process meaning it changes. It changes and it doesn't change in basically the same respect? What's the problem here? It's the "is" notion behind it. If we just say that f(a)=4 and NOT say f(a) "is" 4, meaning the same in every single respect, we won't have such a problem. Therefore, saying f(a) IS A where f(a)=4 gets discarded.

[They are names.]

A function consists of a mapping. A mapping does NOT consists of a name, but a process.

[But that requires it to actually be an operator and maps over a domain.]

No, it doesn't. It can work analogously to ordinary counting.

[You cannot take it and apply it to, say 3+7 since that makes no sense as you used it. What does 3+7 name? 10? It's not in your domain.]

That tells you why you have to specify your domain of discourse.

[Addition isn't even an operator anymore.]

So, what? People used ordinary addition long before it existed as a binary operator such that it maps from the Cartesian product of the reference set and itself to the reference set.

[If it is, and you are using it standardly as over most groups, then you have to include the whole domain of, say, the integers.]

There exists more than one way to do mathematics.

[You don't seem to understand that the operator helps to generate the domain. A set with an operator produces an algebraic system.]

You don't need an operator to generate a domain. Operators may generate a domain, given that we assume closure . But, I can define a universe of discourse such as {a, b, c, d, e, f, g, h, i, j, k} without even having an operator.

[Therefore, (Z,+,0) tells us what the members of Z are because for any member in Z we can use + to find new members in Z because addition is closed. ]

Z qualifies as an infinite set. We can form as many finite sets as we like without even mentioning an operator, because for any finite set S={a1, ..., an} we can always add an extra element an+1 and have S'={a1, ..., an+1}.

[If you take that away, then you have to accept it as either arbitrarily meaningless or not an operator.]

No, counting has plenty of meaning. Almost the whole of human and animal history will disagree with you here if they understood what you mean.

[If your domain is somehow limited to those three elements and it happens to work as you want to define it on one element such that you can name 3+4 as 7, it doesn't mean you can even talk about 7+3 since that doesn't name anything.]

Sure, but you don't have to assume '+' as a binary operator which has closure. Again, arithmetical problems got solved long before this and still get solved by people without a notion of a binary operator on a set.

Good point about the historical gaff I made, but still 0 didn't exist for the Babylonians or Egyptians and they solved arithmetical problems without it.

[what about the subsethood interpretation?

What about it?]

I don't see how the SEP article addresses it.

@bryangoodrich - 

See above.

@Spoonwood - 

Not if a mathematician finds such meaningful.

Yes, because it is meaningful to find instances of when a theorem holds when the values have no meaning whatsoever? That is the point. We apply these theorems in applied mathematics to check them that they actually do represent what they are supposed to.

No I haven't, but it's well understood, so that doesn't make for a relevant criticism.

More like "No, I can't." Sure, summation is not a difficult concept, but you are trying to apply it as if there is some inherent meaning in denotations. Names are meaningless. It is the semantics behind them that provide the meaning. Mathematics formalizes the relations of those intensions so that we can construct a rigorous language to represent those relations. But apparently all that matters is applying arbitrary concepts to arbitrary extensions, since that is all you have done.

It shows that x=2+2 does not exist in R.

And it shows + on R cannot be utilized since wtf is 2? Are they numbers? from what numeric system? Or are they numerals that we are just giving as arbitrary names? That seems to be the case, but you see them with values. You don't get to have your cake and eat it too, here. Either they are just names you can throw around as you like or they derive from a numeric system for which addition is not properly defined. Appealing to "ordinary addition" isn't saying anything whatsoever. You might as well say R={$,#,^} and $+#=^ but #+^ doesn't belong to R. It holds just the same meaning in extension as what you say when you tell me x=2+2 doesn't exist in R. In fact, it shows your + is just something you cannot apply to R. Where does + come from? Once again, define it and you might see where your assumptions are being made.

I don't accept your feigning that you lack an intuitive understanding of what "ordinary addition" means

I understand that we can apply summation over some system of counting numbers for any kind of enumerable set. Let me say R={1,2,3} and I partially enumerate it by f such that f(1)=0, f(2)=1, f(3)=2 and I close our operator so that we are working in Z_3. Then we can demonstrate an easy homomorphism (go ahead if you want) such that f(x+y)=f(x)+f(y) for any x,y in R. In the case of 1+1=1 in R because 0+0=1 in Z_3. 1+2=2, 2+3=1, 2+2=3, etc. A table of this system would work out more visually appealing. The point is that our intuition of addition is captured as applied over certain domains and as a properly closed binary operator, in this instance. You talk like ordinary addition is some universal concept. What if it is addition as applied to clocks? We don't say 12+1=13 but it doesn't exist in our clock_set, we say it equals 1 because we can properly map it to Z_12. When we do it over other kinds of concepts it might work out as simply dealing with N. It varies depending on the context and that is what mathematics has provided us over the years, how to formally understand that. You have completely rejected it by what you've stated here.

All of that comes as absurd, since NONE of those concepts existed for the ancient Babylonians, or the ancient Egyptians.

Yeah, and there were intuitions about how to solve some major classical geometry problems well before we developed field extensions, but that doesn't mean they must have been thinking about field extensions. That would be impossible. The point is that modern mathematics has ironed out the loose, vague and ambiguous concepts of old to prove properly the relations involved. And I don't see how the hell you can say some modular group isn't ordinary addition because it functions EXACTLY THE SAME. The reason it differs has to do with how we set up our domain and establish our binary operator on that system. The clock example is using summation just the same as one sums counting beans. The difference is that the range of his enumeration isn't limited to 12, while the clock's is, in this case.

In order to define it as it's being used I would have to define it in such a way that indicates how an ordinary brain works when it sums together two objects. Set theory does NOT capture this, since a binary process does NOT take place when you add 1+1 or 156+567 in the ordinary sense...

So there are numbers in my head? Or are they numerals? So there is something going on in my head that is doing this process? Because I would really like to know how ordinary addition depends on the mind. Set theory isn't capturing our mental processes. It's not computationally dependent on human cognitive processes, nor humanity at all. It captures relations between concepts and concepts do not exist in the mind or something, unless you want to play the dualist or platonist card, to which I can shape my response to dismantle those failed perspectives.

You may fail to understand what I mean formally here, but I don't need to explain such formally, since you already understand ordinary addition at a much, much more intuitive level.

And if you understood the formality behind it, you would see my point. But that is my further point, you don't understand the formalism involved for wtf one picks up when they count objects in the world. The first chapter of Computability and Logic might help, for instance. I'm not here to educate you on basic things you should pick up on your own, especially if you want to concentrate on mathematics. But it seems more the case you blow off the most fundamental things you need to know.

Do you want to say you don't know how to count?

I know how to count, what you don't seem to realize is that your "set of pants" isn't limited since we can still sum what is not there. If all we have are three pants and we say "take two pants and add two more" wouldn't make sense if we physically don't have more in our collection. But a set is NOT physical. We can still do 2+2 just fine. In fact, what mathematics is capturing is the fact that counting numbers are an infinite set as specified under an operation of addition. It seems the one who is saying they cannot count is you. In fact, you still don't seem to grasp you aren't dealing with NUMBERS in reality. Numbers are a RELATION. If you only have three pants then you've not added 1+1 or any such numeric representation. If you are counting pants then all you can do is enumerate your set of objects. I have already specified what that would entail above. If all you are doing is enumerating your set, you are not really even counting, you're providing a homomorphic mapping from the set to an enumerated set, partial or total. If you try to specify this summation, then either you are dealing with N or some partial listing of it to which you would still need to spell out how counting behaves on this set. If the set is, say, counting hours on a clock, then it DOES act like Z_12 and that IS "ordinary." If it is counting a possibly infinite (conceptually) set of objects, e.g., pants, then you are dealing with N. But who cares about that stuff, right? Just assume it away for some vague unsupported concept about addition and minds and ordinary things that doesn't even make sense to our formal systems. I guess mathematics must have just completely missed the mark all these centuries, right?

In such a case, 2+2 doesn't work as having any solution.

Really? Sounds pretty contrived actually. You cannot count 2 away from 2? What does that mean? Unless you bind it to the set in some meaningful way it is MEANINGLESS. To say we can do summation as "y away from x" is x+y still needs to specify what x and y are. If they belong to some limited set, say Z_3={0,1,2}, then either 2+1 is in the set or your statement about "y away from x" is rather meaningless. But that is not what you say. Instead, you say it is OUTSIDE of the set. So "y away from x in Z_3 is outside of Z_3." You actually want to pass that nonsense off as meaningful?

No. 3+4 indicates two objects with a function operating on them (or something adding them if we don't have the notion of function yet). 7 indicates one object.

Thanks for proving my point again. I assume you also think f(x1,x2,...,xn)=y is saying n+1 different things, right? It's saying we have n x members and a y element and the f statement is ... what? Take an upper division math course already, you can use it if you want to get a handle on this stuff. f(x) DENOTES y. The mapping RELATES the x to the y from one set to another. In the case of a binary operation (being closed) maps it to itself. When you say +(1,2) or 1+2 you are not saying you have three things stated, 1, 2 and 3. You are listing ONE ELEMENT. 1+2 IS the name of 3. You can denote any damn thing numerous ways. 7 is 7, 7 is 6+1, 7 is 5+2, ... and they all mean the same thing. Names are only instilled with the values we construct for them. This isn't even a constructivist argument, either. Even Platonist understand this fact (they just believe there are actual objects in some mathematical realm where numerals do have inherent values and are not merely names. They are more than just the relations they express). To make it even clearer, we will assume my equality statements hold true, then:

f(3+2)=f(3)+f(2)=x+y=z. How many things are stated? ONE. We stated z. It can be named as any of those things. It IS x+y, it IS f(3)+f(2), it IS f(3+2). Your trying to say we have two elements being operated on as if they are separate from the other end of their equality is as bad as saying elements of a set are free on their own without being named a member in a set, as if membership is somehow disjoint from sets. I don't know if that is what those fuzzy logic books are teaching you, but if it is, then it's not mathematics.

A function consists of a mapping. A mapping does NOT consists of a name, but a process.

There's a process in mathematics? I mean, I know there's a process in the metalanguage of mathematics. We can talk about processes in how we solve problems in mathematics, sure. But a process in mathematics? You are wrong again. The mapping denotes a relationship held between the members of two sets. For some a in A and b in B and f:A onto B, such that f(a)=b, f(a) NAMES b. b NAMES f(a). the inverse mapping, let us call it g, such that g(b)=a shows that g(b) NAMES a. Try not to fill the gaps of your lacking mathematical background with poor intuitions about how concepts relate to things in the world. We use those real-world examples to build intuitive understandings of the relations, but the mathematics is abstracted from that. A background in foundations and algebra would help you along in really getting a grasp on what you want to learn in mathematics. A comprehensive study of logic would be beneficial, too. I highly suggest the Burgess book.

1/3=3/9 sure, but 1/3 IS NOT 3/9.

And last time I checked a pizza or any physical object is not a god damn number. Wtf do you think a number is? An object? Do you think it is a first-order individual named of something in the external world? They're not. Frege gives a nice argument of that on his SEP page where he defines two-ness in first-order logic. I am not going to recreate it here.

No, it doesn't. It can work analogously to ordinary counting.

What is ordinary counting? I'm still waiting for your definition. Nevertheless, I don't care if you're counting straw in a barn, you still have a domain specified, and if you are equating addition to enumeration of this set you have a problem. Only educating yourself will help it, and I've offered all I can on that.

That tells you why you have to specify your domain of discourse.

There is more to it. It is assumed you already have a domain since you must be investigating something. But if you're trying to capture the addition on this domain it doesn't do much if you're using some vague concept of summation and then saying your domain is limited such that addition will not be closed. That statement is nonsense since you cannot count what cannot be counted and it isn't a "partial count." You might have a partial enumeration, but like I said, don't convolute the two.

But, I can define a universe of discourse such as {a, b, c, d, e, f, g, h, i, j, k} without even having an operator.

Yeah, but you cannot pull any kind of operator out of your ass. You cannot say "I'm just using ordinary addition." That doesn't mean anything AT ALL. What is a+b?! To even say "I'm adding one pair of pants to another pair of pants" already assumes you enumerated the set! You are dealing with N and representing it by numerals. But the abstract case completely destroys what you are suggesting unless you can tell me you're going to have some obscure notion of mind about which "ordinary addition" is going to tell me wtf a+b means for that set. But you can do it, very easily in fact. Define an enumerating function such that f(a)=0, f(b)=1, ..., f(k)=11 and then do "ordinary addition." But then you're only obtaining that addition over the fact you are using it on a list of positive integers P for which we will say, for it to be an operator, it closes (otherwise you might as well throw logical consequences out the window and theories no longer need to be closed). We can then say our domain is isomorphic to Z_11. In fact, we have the algebraic system (Z_3,+,a) where a is the identity of this group (because it satisfies all the group properties so we call it what it is. We can then conclude nifty things like a+b=b and b+c=d and k+b=a. But fuck all that, let's just say we're using "ordinary addition" and a+b is ... well, I don't know. Maybe the process or the brain will tell us.

I don't see how the SEP article addresses it.

I didn't say it said anything about subsethood. I was talking about probabilities at the time.

@bryangoodrich - 

[Yes, because it is meaningful to find instances of when a theorem holds when the values have no meaning whatsoever? ]

Yes, that can work out as meaningful even if the values have no meaning, because the process of reasoning may have meaning. Or the form involved may have meaning.

[Sure, summation is not a difficult concept, but you are trying to apply it as if there is some inherent meaning in denotations. Names are meaningless.]

Names are not meaningless if you apply meaning to names.

[Mathematics formalizes the relations of those intensions so that we can construct a rigorous language to represent those relations.]

No, it doesn't. Wikipedia define intention as "Intension refers to the set of all possible things a word or phrase could describe." Set theory (crisp) does NOT formalize the relation of the set of all things that the term "set" described. One can legitimatelly call a group of green trees a set of trees. Set theory utterly fails in trying to formalize the intension of the term set in this respect. We may say that it comes close enough to succeeding when we talk about sets which have members that do or do not belong to them. But, for the real-world application here... the concept of "set" and its use in everday language... set theory fails in a ridiculous number of instances.

[And it shows + on R cannot be utilized since wtf is 2? Are they numbers? from what numeric system? Or are they numerals that we are just giving as arbitrary names?]

With R={1, 2, 3} 2 indicates the secound counting number.

[You don't get to have your cake and eat it too, here.]

You can always have part of the cake and still have part of the cake left over.

[Either they are just names you can throw around as you like or they derive from a numeric system for which addition is not properly defined.]

Which standard of definition? I didn't claim to use the standard of definition of modern formal mathematics.

[Where does + come from? Once again, define it and you might see where your assumptions are being made.]

I define it on a finite subset of Z as a process analogous to counting.

[Let me say R={1,2,3} and I partially enumerate it by f such that f(1)=0, f(2)=1, f(3)=2 and I close our operator so that we are working in Z_3. Then we can demonstrate an easy homomorphism (go ahead if you want) such that f(x+y)=f(x)+f(y) for any x,y in R.]

This comes as irrelevant to what I did otherwise, but what the heck.

f(1+1)=f(2)=1, f(1)+f(1)=0+0=0. Huh??? f(1+1) does not equal f(1)+f(1). Oh wait... you've changed the reference set from R={1, 2, 3} to Y={0, 1, 2} since you've specified Z_3 the integers... meaning {..., -1, 0, 1, ...} modulo 3. Since, we have 0 as the identity, this means we have the modulo operation mapping to {0, 1, 2}, so you've switched reference sets on me and that's why the above didn't work out. Instead of swapping R={1, 2, 3} to Y={0, 1, 2} we can say {1, 2, 3} as our reference set and say we work in a structure isomorphic to Z_3 where we have 1+1=2, 1+2=3, 1+3=1, 2+2=2, 2+3=3, 3+3=3. I'll switch the reference set from {1, 2, 3} to {0, 1, 2}. Actually, I can't do that either since you have 4 elements 0, 1, 2, and 3 in the definition of f "f(1)=0, f(2)=1, f(3)=2". So, I guess I'll specify the reference set as {0, 1, 2, 3}. Then, we have

f(1+1)=f(2)=1, f(1)+f(1)=0+0=0
f(1+2)=f(0)= oh drat... you defined addition as operating in Z_3 and didn't define f(0). Sorry, no homorphism. I feel sure you got very close to getting one here.

[In the case of 1+1=1 in R because 0+0=1 in Z_3.]

Huh, 0+0=0 in Z_3.

[1+2=2, 2+3=1, 2+2=3, etc.]

Huh? 2+3=1 if we have 1 as our identity and the reference set {1, 2, 3} with an isomorphism to {0, 1, 2} via the relation of addition mod 3 on {0, 1, 2} with the refernence set of the modulus as N. Since 2+3=1, 2+2=2, as 1+2=0, and 2+2=1 in Z_3.

For yours to work properly (in the modern sense) we first have to specify our reference set as Z. Then, we can say that f consists of a (closed) unary operation such it maps from Z to Z. In other words f:Z->Z. Then, we specify the behavior of our function as f(1)=0, f(2)=1, f(3)=2. Why can't we just have 3=0 since that happens in Z_3 for the reference set Z? Because we didn't specify the existence of such an element 3 as permissible for the relam of our discussion. For the purposes of speaking, 3 has no existence whatsoever, while 0 does have existence. Since it has no existence in a reference set such as {0, 1, 2}=J, if we specified J as our reference set, f:J->J does not exist as 3 does not belong to J. You can specify Z_3 for addition, but then we have antoher operator '+' which acts on Z, which means we have another reference set at work for the operator '+' than we have for f. The simple solution comes as just to specify our reference set as Z as indicated above. Now, does f(x+y)=f(x)+f(y) hold for the reference set Z, where '+' operates commutatively on the subset {1, 2, 3} of our reference set Z isomorphic to '+' on {0, 1, 2} by which I specifically mean 1+1=1, 1+2=2, 1+3=3, 2+2=1, 2+3=2, 3+3=3, for the f:f(1)=0, f(2)=1, f(3)=2,?

f(1)=0, f(2)=1, f(3)=2

f(1+1)=f(1)=0, f(1)+f(1)=0+0=3+3=3=0 (you have to convert each 0 to 3 individually, which you can now do since you have your reference set as Z, you can't convert 0+0 to anything in one step though, since we've defined '+' on the reference subset {0, 1, 2}.)
f(1+2)=f(2)=1, f(1)+f(2)=0+1=3+1=1+3=3=0. Oh, no homomorphism.

Did I misunderstand your problem? I tried to adjust it so it would have a homomorphism, but unless I did my calculations wrong, it failed. If you want to go back to what you said here "Let me say R={1,2,3} and I partially enumerate it by f such that f(1)=0, f(2)=1, f(3)=2 and I close our operator so that we are working in Z_3." Remember I defined R as a reference set . For a function to operate in a closed manner it has to map from the universal set to the universal set. For this to happen, in specific instances, the function has to map an element a in the universal set X to an element b in the universal set. If not, then we can't write f:X->X. You don't have to assume a function as operating in a closed manner though, since the characteristic function opeartes outside the universe of discourse. So, the real problem here becomes saying "Let me say R={1,2,3} and I partially enumerate it by f such that f(1)=0, f(2)=1, f(3)=2 and I close our operator so that we are working in Z_3." doens't work, because the sub-statements "R={1,2,3} and I partially enumerate it by f such that f(1)=0, f(2)=1, f(3)=2," and "I close our operator so that we are working in Z_3," can't work in conjunction since 0 does not belong to R. If you have your way around this and have your intended homorphism, I'd like to see the proof.

f(1)=3, f(2)=1, f(3)=2, 1+x=1, 2+2=1, 2+3=2, 3+3=3, a+b=b+a for R={1, 2, 3}

f(1+1)=f(1)=3, f(1)+f(1)=3+3=3
f(1+2)=f(1)=3, f(1)+f(2)=3+1=3
f(1+3)=f(1)=3, f(1)+f(3)=3+2=2+3=3
f(2+1)=f(1)=3, f(2)+f(1)=1+3=1
f(2+2)=f(1)=3, f(2)+f(2)=1+1=1... drat that didn't work either.

[What if it is addition as applied to clocks?]

It's not ordinary addition.

[We don't say 12+1=13 but it doesn't exist in our clock_set...]

People do do so. Computers do so, and the military has for years. The statement 12+1(mod 12)=1 indicates that for ordinary addition, which has no modulus, we have 13 as the right part of the equation.

[we say it equals 1 because we can properly map it to Z_12.]

No, getting one a clock works as more intuitive and less formal than that. We define addition on a clock as succession in a certain direction around the clock which we know as 'clockwise'. Since 1 comes 1 hour unit after 12, 12+1=1 on a standard clock. There does exist an identity, but we don't have a group structure on an accurate clock or one we don't manually adjust the time, since we don't have a reverse direction of time... the hand only moves clockwise. So, for addition on a clock we really only have a commutative monoid.

[It varies depending on the context and that is what mathematics has provided us over the years, how to formally understand that. You have completely rejected it by what you've stated here.]

No, because the reference set provides a context to what we do. What 2+2 means depends on the context still, since 2+2 equals 4 in one context and doesn't equal anything in another.

[The point is that modern mathematics has ironed out the loose, vague and ambiguous concepts of old to prove properly the relations involved.]

According to its own standards of precision, maybe. Plenty of texts though indicate that difficulties like the paradoxes of intuitive set theory still exist.

[And I don't see how the hell you can say some modular group isn't ordinary addition because it functions EXACTLY THE SAME.]

No, it doesn't. For modular addition on the ordered set {1, 2, 3} with the order 12.

[The clock example is using summation just the same as one sums counting beans. ]

Children learn addition, or at least that's what I think I've read and how I remember it, in terms of counting. With counting, we have a precedence relationship 1<2<3 and so on. We also have a greater than and a less than relatioship for counting, 1<2<3 and so on. For clocks we don't have such a greater than relationship, since 1 O'Clock P.M. is not greater than 12 O'Clock A.M.

[Because I would really like to know how ordinary addition depends on the mind.]

Suppose not. Then, where does ordinary addition exist? It can't exist in definitions, since definitions depend on a mind/brain to understand them. So, in the real world independent of our minds? No, not there either. So, in the mind.

[Set theory isn't capturing our mental processes. It's not computationally dependent on human cognitive processes, nor humanity at all.]

So, without any mind/brain whatsoever, set theory still exists? I certainly don't agree with that.

[It captures relations between concepts and concepts do not exist in the mind or something, unless you want to play the dualist or platonist card, to which I can shape my response to dismantle those failed perspectives.]

Concepts, by definition, consist of what the mind does when thinking. So, they exist in the mind. Hey look, dictionary dot com says "2. Something formed in the mind; a thought or notion. See Synonyms at idea." Second, there exists a more grave problem here. Philosophical ideas don't really "fail" like that. They might not come as so promeninent anymore, but they stick around. Pick a political persuasion you don't like, and speak of it in terms of its philosophical ideas. Well, whatever you pick, that political persuasion still exists, so it really hasn't failed. Maybe its dumb, but it still exists, so to say "its failed" comes as another matter, since it still has enough a degree of success to still exist.

[And if you understood the formality behind it, you would see my point.]

Yeah, yeah... claim I don't understand it. So you can make an ad hominem, so what?

[But that is my further point, you don't understand the formalism involved for wtf one picks up when they count objects in the world.]

People counted objects before ANY formalism existed. People still do today, especially those in aboriginal cultures. So, talking about "the formalism" doesn't make sense.

[The first chapter of Computability and Logic might help, for instance.]

I seriously don't know what I bother with this. The indicator function proof got interesting, I wish you would have commented something on that. But, when you forget that people counted before any sort of writing even existed and counted before any formalism, and you talk about a "closed" function on a reference set J where it can't work out as closed since we have it mapping outside J, and you insist I made a mistake by saying the empty set is not a member of any set, and you say Russell's paradox talks about a set of sets which is not a subset of itself, instead of a set of sets which is not a member of itself and then you say "I'm not here to educate you on basic things you should pick up on your own, especially if you want to concentrate on mathematics." how can I respond with anything but "REMOVE THE MOTE FROM YOUR OWN EYE."

[But it seems more the case you blow off the most fundamental things you need to know.]

Like what? How closure works? That sets work as collections of members instead of saying "sets are the members"? That 'e' means 'member' and not subset? That counting existed long before writing and thus long before formal systems? That when you add positive numbers a+b in ordinary addition you always get a result c, such that c>a, and c>b?

[I know how to count, what you don't seem to realize is that your "set of pants" isn't limited since we can still sum what is not there.]

The "set of pants" works out as finite in that there only exist so many pants in the world. When adding numbers of REAL pants together, you can't add what doesn't exist there, because if it doesn't exist there, it doesn't qualify as REAL (at present at least).

[But a set is NOT physical.]

You can consider sets as physical. It doesn't generally work out so well, but in some cases it works.

[In fact, what mathematics is capturing is the fact that counting numbers are an infinite set as specified under an operation of addition.]

In our culture. But, for an aborigine who has {1, 2, 3, 4, 5, MANY} as his counting system, the numbers used to count don't qualify as an infinite set.

[In fact, you still don't seem to grasp you aren't dealing with NUMBERS in reality.]

According your and perhaps many mathematicians unordinary definition of "numbers".

[If the set is, say, counting hours on a clock, then it DOES act like Z_12 and that IS "ordinary." ]

c>a and c>b holds for a+b in ordinary addition, but not for a clock.

[Just assume it away for some vague unsupported concept about addition and minds and ordinary things that doesn't even make sense to our formal systems.]

At best those formal systems come out as limited to planet Earth and the culutre of planet Earth. Do you want to claim that they hold as supported in some sort of universal sense throughout the universe and no organism in our local group of galaxies has some sort of mathematical system more rigorous and more formal than ours? Do you think there simply COULD NOT exist such a system which makes our modern mathematics look as informal, loose, and "vague" as some modern mathematicians view Euclid's system? Or makes our mathematics look as primitive as we see the mathematics of the ancient Egpytians? If you claim our formal systems as that darn rigorous and that darn formal in effect you do that. If you claim formality and rigor as matters of degree, and claim some modern mathematical systems as formal and rigorous, in effect, you do that, since such formal and rigor holds universally. But, to say that comes as the pitch of arrogance. Rigor and formality come in degrees.

[Really? Sounds pretty contrived actually.]

All of mathematics works out as contrived since it doesn't exist anywhere in the external world, but merely models it to some degree, so what?

[You cannot count 2 away from 2?]

You can't count 2 away from 2 in {1, 2, 3} just as you can't solve i^2=-1 in the real numbers for i.

[Unless you bind it to the set in some meaningful way it is MEANINGLESS.]

But, since we have {1, 2, 3} as our set 1+1=2 and we can count 1 away from 1. So, '+' has a meaning in at least one instance. It also has meaning in 1+2 and 2+1, so assuming no commutavity for '+', it has meaning in three instances. It doesn't have meaning for 1+3, 3+1, 3+2, 2+3, and 2+2, so in 5 instances '+' doesn't have meaning. So, '+' has degree of meaning of 3/8, assuming degree of meaning for a relation r equals number of instances in which it has a meaning/number of total instances possible for r to happen. We have a degree of meaning greater than 0, so it doesn't work out as meaningless.

[If they belong to some limited set, say Z_3={0,1,2}, then either 2+1 is in the set or your statement about "y away from x" is rather meaningless.]

You've switched the reference set from {0, 1, 2} to Z_3={0, 1, 2}, since the reference set I used did not assume a function with a modulus, while yours did. By default, if no function or modulus works in Z and you specify a subset of Z, you mean that subset as your reference set as it would behave as if it existed in the superset Z. You've changed its behavior, since you've said Z_3. Second, if you want to now make an anlogy with clocks, you'll have to specify how you compute "away"... in other words a metric. Why? Well, we can always move across a clock to measure the distance between two hands. In ordinary addition you don't have to do this, because you don't have to make an analogy with clocks.

[So "y away from x in Z_3 is outside of Z_3."]

No, I didn't say that, becuase I didn't specify Z_3 as my system.

[I assume you also think f(x1,x2,...,xn)=y is saying n+1 different things, right?]

No, more.

[f(x) DENOTES y.]

Even supposing so, denotation does not mean "is the same as" since connotation comes as another way to define things.

[When you say +(1,2) or 1+2 you are not saying you have three things stated, 1, 2 and 3.]

You have three "things", 1, 2, and '+'.

[You are listing ONE ELEMENT.]

You list an equation that equals an element.

[1+2 IS the name of 3.]

Not on the reference set {3}. In such a case 1+2 doesn't make any sense.

[7 is 7, 7 is 6+1, 7 is 5+2, ... and they all mean the same thing.]

No, 5+2 in Z_6 equals 1... not 7, because no 7 exists in Z_6.

[f(3+2)=f(3)+f(2)=x+y=z. How many things are stated? ONE.]

No, you have complete sentences there with a verb '=', a noun 'f'... look I can translate your sentence as
A function of 3 plus 2 equals a function of 3 plus a function of 2 which equals x plus y which equals z. Equality does NOT mean sameness, since the result of counting my toes may equal the result of counting my fingers in terms of a number. But, I have NOT done exactly the same thing since the action of counting on my toes than on my fingers differs in that I work with different things when counting and look at different objects when I count. Since I look at different objects, my brain has different information when it counts. Since it has different information, the process of counting does NOT work out as exactly the same in both cases... my brain has to sort through the information in slightly different ways... it has to adjust what it recognizes as permissible object to count.

[It IS x+y, it IS f(3)+f(2), it IS f(3+2).]

No, you think about 2+1 differently than 3. And mathematical objects come as concepts.

[Your trying to say we have two elements being operated on as if they are separate from the other end of their equality is as bad as saying elements of a set are free on their own without being named a member in a set, as if membership is somehow disjoint from sets.]

So your subjective evaluation says, so what?

[I don't know if that is what those fuzzy logic books are teaching you, but if it is, then it's not mathematics.]

By what definition? According to who's decree? Maybe they imply that, that 'is' and 'equality' mean different things, but I really get that from reading General Semantics ideas. Zadeh once said something to the effect that GS and fuzzy logic have a lot in common... I don't know if he meant this sort of thing though.

[There's a process in mathematics?]

Mathematics itself consists of a process, so yes.

[For some a in A and b in B and f:A onto B, such that f(a)=b, f(a) NAMES b.]

No, f(a)=b. Equality does not consist of naming the same thing... it's not sameness.

[Try not to fill the gaps of your lacking mathematical background with poor intuitions about how concepts relate to things in the world.]

What's in the world here? Our function f(a), the concept of a function? Shared concepts of ours?

[Wtf do you think a number is?]

A concept.

[What is ordinary counting? I'm still waiting for your definition.]

You understand how to do it without a definition and knew how to do it without a definition before the age of 10. So, you can still tell me if 2+2, where '+' indicates a process analogous with counting, exists in our given reference set {1, 2, 3}. Asking for a definition comes as irrelevant, since you can still how doing counting in such a situation works, and when it doesn't work.

[Nevertheless, I don't care if you're counting straw in a barn, you still have a domain specified, and if you are equating addition to enumeration of this set you have a problem.]

You can add or enumerate over a reference set such as {1, 2, 3}. I don't care... say we want to think of '+' as enurmeration. Fine. Does {x: 2+2=x} always give the same answer given no reference set? No, because on {1, 2, 3} (again, no Z_N specified, so our reference set works as a subset of Z) yields {}, while on {1, 2, 3, 4, 5} we have {4}. Oh, of course. Now I see how to satisfy your desire for closure. Or at least something more close to it.

Suppose we want to write {x: 2+2=x} in specific terms of a member of our refence set. I'll define our reference set as K={1, 2, 3, {}}. '+' may not work out a closed binary operator exactly, but {x: 2+2=x} always works out as a subset of K. In particular
{x: 2+2=x} has no x which belongs to K as its solution, so we have {x: 2+2=x}={}. {} belongs to K, so we still have something *like* closure and can rather easily construct something similar for a finite set.

[But if you're trying to capture the addition on this domain it doesn't do much if you're using some vague concept of summation and then saying your domain is limited such that addition will not be closed.]

See above.

[That statement is nonsense since you cannot count what cannot be counted and it isn't a "partial count." You might have a partial enumeration, but like I said, don't convolute the two.]

Um... I don't know what you mean exactly, but there exists the notion of fuzzy cardinality, which can give fractions for the cardinality. For the fuzzy number 3={(1, .4), (2, .8), {3, 1), (4, .8), (5, .4)} we have a sigma count or cardinality equal to the sum or integral of the membership functions, or .4+.8+1+.8+.4=3.4 So, some sort of "partial counting" can get said to get involved somewhere if you view fuzzy sets as "sort of" belonging to their reference set.

[Yeah, but you cannot pull any kind of operator out of your ass.]

You can add the empty set to any reference set and then do something like what I did above for something like an operation.

[What is a+b?! To even say "I'm adding one pair of pants to another pair of pants" already assumes you enumerated the set! You are dealing with N and representing it by numerals. But the abstract case completely destroys what you are suggesting unless you can tell me you're going to have some obscure notion of mind about which "ordinary addition" is going to tell me wtf a+b means for that set.]

a+b without instantation doesn't exist in ordinary addition. I don't have to tell you what it means in ordinary addition, because alegbraic symbolism like a+b doesn't exist in ordinary addition.

[(otherwise you might as well throw logical consequences out the window and theories no longer need to be closed). ]

I don't see how theories need to work as closed, nor how without them they won't have logical consequences. Also, as indicated you can pretty much always get something like closure via the empty set if you like.

[But fuck all that, let's just say we're using "ordinary addition" and a+b is ... well, I don't know. Maybe the process or the brain will tell us.]

The set of {a+b} equals the empty set {}. So, we need another system if we want a solution set with a member.

[I don't see how the SEP article addresses it.

I didn't say it said anything about subsethood. I was talking about probabilities at the time.]

I don't see how the SEP article addresses the subsethood view of probability.

@bryangoodrich - 

Edit: For problems like the {x:x=2+2} problem above on just set like F={1, 2, 3, {}}, either we have a solution which equals a member of F, or we have a solution set with a member of F. E.G. {x:x=2+2} equals {}, a member of F. {x:x=1+1} equals {2} a solution which which has 2, a member of F as its solution.

Note, you might want to start with the closing comments at the bottom.

@Spoonwood - 

Yes, that can work out as meaningful even if the values have no meaning, because the process of reasoning may have meaning. Or the form involved may have meaning.

I was referring to semantical meaning, not some loose poetic idea you are appealing to.

Names are not meaningless if you apply meaning to names.

Wow ... great insight. Then we're not talking about mere denotations anymore but the semantics behind it. That was the point.

set theory fails in a ridiculous number of instances.

It fails what exactly? To be an exact model of reality? If that were possible then it would say reality behaved exactly like set theoretical concepts. No one in their right mind things an object language is the metalanguage. It represents it in a formal manner to better improve our understanding of relations in the metalanguage.

With R={1, 2, 3} 2 indicates the secound counting number.

Does it? You first need to provide that enumeration. Say f(r) for r as members of R such that f(1)=1, f(2)=2 and f(3)=3. But what are we dealing with was my point. Numbers (things with meaning in some context to a numerical system) or numerals? As the author states in Boolos, et. al, Computability and Logic 5th ed. (New York: Cambridge University Press, 2007), pp. 3-4, "Note that entries in these lists are not numbers but numerals, or names of numbers. In general, in listing the members of a set you manipulate names, not the things named." They go on, "But we might also speak of the entries in the list as names of themselves so as to be able to continue to insist that in enumerating a set, it is names of members of the set that are arranged in a list." Of course, we can just assume they are ordered in such an enumeration as listed (an acceptable convention), but you missed the entire point I was making that you don't realize the difference between extension and intension.

I define it on a finite subset of Z as a process analogous to counting.

Well it cannot be the operator from Z because one of the first things you learn in algebra is that a subset of a group takes the operator with it to form a subgroup. So we should be in agreement no operation is being talked about. So you revert "ordinary addition" to counting to which you seem to be mistaking enumerating a set as actually performing an operation? Well, it isn't. All you are saying is I can count this set up to three members, it is a partial enumeration. All you are really saying then is "I cannot enumerate the set larger then the set can be enumerated" if you're saying you take some enumerated value of the nth place and add an mth amount that goes beyond the partial listing. But you are not performing an operation on numbers, you are taking about names of them. If you were performing addition, some kind of summation, doing an operation on a set, then it would be like the examples I expressed before.

This comes as irrelevant to what I did otherwise, but what the heck.

So then the examples I offered are irrelevant and you're stating a pointless tautology that you cannot enumerate a set further than its partial listing is enumerated. Brilliant! But my point is not to mock you but to demonstrate the fact you don't realize the content of what you are talking about, which is why I have offered the literary suggestions, that can be found at many libraries or at least in previews from google. The concepts of mathematical foundations and logic are well developed and can be found in numerous places, but required for a strong logical background is a grasp of computability theory (enumerability, uncomputability, recursion and usually Godel's proofs), as well as metalogic (nth-order logics (types), models, proof theory or provability, and undecidability). I don't even have the complete (formal training, my college only went over loose syntax and concepts, graduate school requires mathematical rigor and precise understanding of these things). That is why I bought the Boolos book (paper back is around 20 bucks, damn well worth it, and for the 5th, recent, edition!).

Anyway, as for the homomorphism, I enumerated it backward. A guys-night-in can cause that. I should enumerate it as f(1)=1, f(2)=2, f(3)=3 (realizing that the list is only names--numerals, and not meaningful numbers). The slight of hand I didn't properly spell out was mapping onto Z_3. Enumerating R as above we have f(n)=n for any n in R. The enumeration of Z_3={0,1,2) is f(n)=n-1 such that f(1)=0, f(2)=1 and f(3)=2. As I stated above, if all you are trying to talk about is enumerating the list, then the partial list of R can be made homomorphic to that of Z_3 since they are homomorphic, such that for any n in R, h(n)=n-1 in Z_3. What does this tell us? The notation is comparing the enumeration of the set (the listing). It is not stating an operation, mind you. Therefore h(1)=0 in Z_3, i.e., h(1) IS 0 in Z_3. h(2)=1 and h(3)=2. The operation in Z_3 tells us the behavior that for n=1 (the first member in the set by its enumeration) we relate 1 in R to the identity as it is in Z_3, and the rest follows by the properties of homomorphisms. But to be clear, a homomorphism would take the operation from R, such that h(x*y)=h(x)+h(y), we have two different operations specified. * is that in R while + is that in the image because h(x) and h(y) ARE those members in the image. If we call them a and b then h(x*y)=a+b, and, in fact, we can call those singular terms (since we're only naming them) such that h(n)=m for x*y IS n in R and m IS a+b in the image. Just to reiterate some facts. But R doesn't have any operations. What you were trying to do was talk about some arbitrary concept of summation when what you really meant, as articulated above, is that of specifying the enumeration. By both appealing to the enumerations of these two sets and a valid homomorphism between them, I have provided that we can spell out an operator on R that is addition in any "ordinary" sense, and it maintains the enumeration. But that is moot since you really aren't talking about operations, you are only talking about enumeration and a trite tautology as spelled out earlier.

Did I misunderstand your problem?

No, I enumerated backward. What I just said does hold, and it is a valid homomorphism if we take R to have a binary operator, such that they are enumerated the same and 1 in R becomes the identity in Z_3. In other words, we can turn R into an algebraic system (R,+,1) or ({1,2,3},+,1) which is isomorphic to the group Z_3, i.e., ({0,1,2},+,0).

It's not ordinary addition.

Really? Because so far you've seem to equated "ordinary addition" to enumerating (listing) a set. And it very well is the case that a clock falls into that scheme as I did it with the Z_3. We can easily do it with Z_12. Try it if you don't think so.

People do do so. Computers do so, and the military has for years. The statement 12+1(mod 12)=1 indicates that for ordinary addition, which has no modulus, we have 13 as the right part of the equation.

For one, computers generally deal with Z_2. The military deals with clocks of a Z_24 kind. And we don't say "13 isn't in our clock set" we say "it's 1 o'clock now" and there is no confusion. You are really stretching if you think enumerating a set like Z_12 is impossible. Oh, but here is why, let us move to a total listing (I've only compared, as in the R to Z+3 example above, partial enumerations). It really isn't hard because f(13)=1, f(14)=2, and so on. This isn't a problem since enumeration can have repeated entries. A valid list of the positive integers can be given by 1,1,2,2,3,3,4,4, ... How does this fact coincide with the previous analysis? Well, if we are going to give a total enumeration of our set R then you need to spell out what f(4) is, and since you seem to think 2+2 just isn't in the set, well, 4 isn't by that name, but we have to count it some how in enumeration, if we give a total enumeration (a complete mapping to the positive integers, a complete enumeration). This is why I picked out Z_3, the isomorphism of R being enumerated. Because f(1)=1, f(2)=2, f(3)=3 and f(4)=1, f(5)=2, f(6)=3. Our listing will clearly be 1 2 3 1 2 3 1 2 3 ... This enumeration is just like enumerating Z_3! Amazing, they are also isomorphic, and if we do define an operator then, it seems to also behave the same. I didn't show an isomorphism, however, just a homomorphism. So to be fair I should spell out the fact that the kernel of this mapping of R to Z_3 is one-to-one, such that the only member mapped to the kernel is the identity of R, which, as should be clear, is 1.

No, because the reference set provides a context to what we do. What 2+2 means depends on the context still, since 2+2 equals 4 in one context and doesn't equal anything in another.

If you are done grasping at straws you might stop and realize that this "2+2" you are talking about as some universal "ordinary addition" is not an operation, it is not even addition. It is specifying the enumerated place in a list! "2 places passed the 2nd place in the list." If you are saying "all we have is a partial enumeration and 2+2 doesn't exist", then that is a trite tautology about having a finite set and list. But that doesn't stop you from being capable of producing a redundant total enumeration which will help us see the obvious intuitive fact that the finite set is isomorphic, in the cases presented, to modulo groups. In this case, the domain may be finite but that doesn't mean we cannot give a complete listing. In fact, to be considered enumerable (countable) it must map to N. By definition, "an enumerable, or countable, set is one whose members can be enumerated: arranged in a single list with a first entry, a second entry, and so on, so that every member of the set appears sooner or later on the list," (Boolos, et al. 2007, p. 3). The wiki pages on countable sets, recursively enumerable sets and recursive sets, might be beneficial, also.

According to its own standards of precision, maybe. Plenty of texts though indicate that difficulties like the paradoxes of intuitive set theory still exist.

Such as? And the standards aren't as arbitrary as seems suggested by what you say. This is because some of what is in logic is related to that of the real world. The standards are influenced by application and reality and, of course, humanity or human minds. In part, this is why mathematics is not divorced from concerns of epistemology and metaphysics.

No, it doesn't. For modular addition on the ordered set {1, 2, 3} with the order 12.

I don't know what you mean by "with the order 12" on the ordered set (1,2,3). Note, the brackets I use for ordered sets. Lighthouse describes that the best way to really depict order through set notation alone would be, in this case {1,{1,2},{1,2,3}), due to the fact that repeated entries in a set are reduced to the singular instance. Nevertheless, I have satisfactorily shown above how "ordinary addition" works on modulo groups just fine.

So, without any mind/brain whatsoever, set theory still exists? I certainly don't agree with that.

Definitely not a platonist then. Sound exactly like a constructivist, actually.

Anyway, I am not going to bother responding to the rest of what you say as much of it I have addressed the fundamental aspects that are relevant. Don't mistake my mocking or declaring your ignorance as an ad hominem, unless you wish to take it personal. You demonstrate a clear lack of understanding of the material. I have shown that lack of understanding numerous times. My stating your ignorance is simply recalling that fact. Do not bother responding if you expect a reply since school is starting and between the six classes, working and doing research and doing whatever writing and website development I can, I will not have much time to commit to back-and-forth correspondence with people. I am changing the intent of my Blog, however. If you have any specific questions then go ahead and send me a list of them and I will quote you and answer, as I instructed EGM in a recent blog. In the question you can specify the length of response you want, whether short or long or leave it up to me. I will get to responding to them as soon as is convenient to be done. Otherwise, I will publish my own stuff and specifics on my website.