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This is more of a philosophy/foundation question.

I usually come across things like "the set of all men", or for example sets of symbols, i.e. sets of non-mathematical objects.

This confuses me, because as I understand it, the only kind of objects that exists in set theory are sets. It doesn't make sense to speak of other objects unless we have formalized them in terms of sets. So what to do with something like the set of all men? Are we working with a different set theory, a naive one? Or is it that we are omitting the formalization, because it is straightforward (e.g. assign a number to every man)?

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This appears to be a question of mathematical modeling en.wikipedia.org/wiki/Mathematical_model, not of set theory per se. Men themselves are not mathematical objects, so it shouldn't bother you that they are not sets. –  Jonas Meyer Mar 18 '10 at 21:12
    
As far as mathematical objects go, this page is quite relevant: en.wikipedia.org/wiki/Formalism_(mathematics) . For example: "the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other contensive subject matter — in fact, they aren't "about" anything at all." So, at least according to this philosophy, it would be meaningless to talk about "the set of all men" - we could write down a letter, say "S", denoting a set whose elements one might choose to interpret as men, but not a set of actual men. –  Zev Chonoles Mar 18 '10 at 21:42
    
Sorry about the link, I can't seem to get it to work. –  Zev Chonoles Mar 18 '10 at 21:44
    
    
Isn't it just a question of "tags"? The "set of all men", whatever that may be has some cardinality k. You might as well be working with any set A with cardinality k, as long as you have a good "meta-mathematical dictionary" to translate anything you say about A in terms of men and viceversa. –  babubba Mar 18 '10 at 22:12

2 Answers 2

You don't have to define your objects as sets, in fact, you should avoid such unnatural definitions. I don't think a number theorist would be happy to see a proof referring to elements of a natural number or using the identity $1=\{0\}$. Such proofs are not acceptable because they won't survive even the slightest change in the foundations.

Similarly, if you develop Euclidean geometry, you don't define a point as a two-element set whose first element is a Dedekind cut and the second one is another weird set. You rather begin with axioms (either Euclidean ones or some axiomatization of the real line) and build the geometry on these. The set theory comes in if you want to show that your theory is consistent (as long as ZF is), and you do that by building a model within ZF.

In your example with men, your actually create a mathematical model of whatever you want to study, in the same way as physics does. There are always translation steps from real world to mathematics and back, they just happen to be trivial in this case. So it's not a problem that men are (modelled by) sets.

Only if you like to believe that mathematical objects do exist in some metaphysical sense, you will have a problem with the counter-intuitive claim that everything is a set. But you can just remove this axiom and stay agnostic about whether everything is a set or not. You will not lose anything - the only essential use of this axiom is that you are able to define the notion of equality rather than having it built into logic. And this is hardly of any importance outside the logic itself.

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If you are being, say, at least semiformal in your approach to set theory, whether or not objects which are not sets exist depends upon the particular brand of set theory you choose. The most common contemporary set theory, ZFC, is a "pure set theory", in which every object is itself a set, so the men indeed do not form a set.

But there are other set theories which allow non set elements, or urelements (what a great name!). In particular, Quine's New Foundations with Urelements is a relatively popular such theory.

So far as I know it is towards the philosophical end of the spectrum to worry about whether sets should be allowed to contain urelements or not. The mathematical justification for this is that, using the axiom of choice, any set can be put in bijection with a von Neumann ordinal, hence a pure set. But you should be able to speak of sets of men if you want to, I suppose.

Addendum: I like Sergei Ivanov's answer. He hits the following key point: if you ask a generic mathematician whether or not an object which is not a set can be an element of a set, you will not get either "yes" or "no" as an answer, but rather an explanation of why they regard the question as being a mathematically unfruitful one. When using sets for mathematical purposes, the "nature" of the objects which comprise sets is now regarded as being completely irrelevant. This is the "structuralist" approach to mathematics, which has been clarified and taken further by the more modern categorical approach.

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Excellent answer Pete! I wish more mainstream mathematicians were so well acquainted with foundational issues. Note that ZFA (ZF with Atoms, aka ZFU) is more popular than NFU. I don't know why wikipedia doesn't mention this theory. The main reason for NFU's "popularity" is that it has been shown consistent (relative to PA!) whereas the consistency of NF is still a wide open problem. With respect to foundations, I think this is by far one of the most unexpected results ever! –  François G. Dorais Mar 18 '10 at 22:28
    
It seems to me,Pete-see my response above in addition-that unless you want to take a purely instrumentalist approach to mathematics,these issues need to be front and center for all of us.That's just my feeling on it. –  Andrew L Mar 18 '10 at 23:38
    
I always understood urelements as atoms or monads: totally propertyless entities, only distinguishable one from each other but not on behalf of their - not existent - properties. Such urelements alone give set theory another flavor. But it might be hard to see men - as opposed to women? - as such urelements. One might argue: something, that has properties, must have something like an inner structure, thus must be based on some kind of set. –  Hans Stricker Mar 19 '10 at 7:10
    
haha, the name urelement can be interpreted as a pun. urelement = your element. I don't know whether it was in the original intention or not, was Quine the one who gave that name? –  Tran Chieu Minh Mar 19 '10 at 7:45
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The name urelement comes from the german-derived prefix ur- (primitive). Then urelement means "primitive element". –  Harry Gindi Mar 19 '10 at 8:43

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