# Mathematical properties not expressible in set-theoretic terms

It was once assumed that all of mathematics could be expressed in set-theoretic terms.

I understand that this is no longer the case.

Can anyone provide examples of mathematical properties which resist set-theoretic representations.

I would especially be interested in properties of numbers - not arithmetic properties, obviously.

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What makes you think that this is no longer the case, if you know of no examples? – Cameron Buie Aug 5 '13 at 2:52
All modern mathematics can be formalized in ZFC or NBG. – Yury Aug 5 '13 at 2:53
@cameron buie. I'm reading philosophy of mathematics, and I have read a number of passing references to non-set-theoretic areas of mathematics. (No jokes about philosophy please) – Nick R Aug 5 '13 at 2:55
@NickR : +1, Also please give some references to philosophy of Math you have – Arjang Aug 5 '13 at 2:58
@Yury: You are right, of course, but one should add the "large cardinals axiom" which, nowdays, is considered a pair of the "standard" axiomatics of the set theory. – studiosus Aug 5 '13 at 2:58

Perhaps what you are referring to is the use of set theory as foundations for mathematics and the existence of other foundations. Other foundations do exist, but that does not mean set theory is insufficient. It's more that simply other foundations exist. Some foundations can be better suited for some purposes and not for others. It depends a lot on what you wish to accomplish.

A recent major advancement in non-set-theoretic foundations of mathematics is homotopy type theory, see here.

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That's an interesting possibility. One of the problems with reading philosophy it that philosophers are rarely able to clearly express their thoughts. – Nick R Aug 5 '13 at 3:20
@Nick R: I thought you said "no jokes about philosophy please"? – Mikhail Katz Aug 6 '13 at 19:48
@user72694 Ok, so "rarely" is a bit strong. However, on occasion I find myself closing a book in order to examine the cover - just to make sure that I am not reading some sort of experimental novel written in a stream of consciousness style. – Nick R Aug 10 '13 at 19:09

There are no number-theoretic examples. Problems arise when one wants to deal with mathematical objects as big as the class of all sets or even bigger yet. Various methods have been developed for bringing such topics into the scope of set theory (reflection principles and Grothendieck universes are two such methods), but one can argue that they do not literally deal with the class of all sets or similar classes directly but rather provide a set-theoretic surrogate with the same essential properties.

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Andreas, the way you phrase it, how is this not true of essentially everything? The natural numbers, one can argue, are not literally $\omega$, but this is rather a set-theoretic surrogate, etc. – Andrés E. Caicedo Aug 5 '13 at 3:01
Andreas, I suspect that you are correct regarding number-theoretic examples. Proper classes, while not sets, are represented in set-theoretic terms. Similarly, the existence of large cardinals cannot be proven in ZF, but they are describable in set-theoretic terms. – Nick R Aug 5 '13 at 3:09
@NickR "Set-theoretic" is not the same as "in $\mathsf{ZF}$", or "in $\mathsf{ZFC}$", or "in $\mathsf{ZFC}+$there is a proper class of inaccessible cardinals" or... – Andrés E. Caicedo Aug 5 '13 at 3:11
@AndresCaicedo One difference is that, when we replace the class $V$ of all sets by a Grothendieck universe (or a sufficiently reflecting $V_\kappa$), the replacement merely has, as I wrote, the same essential properties. In the case of the intuitive natural numbers and their von Neumann codes, the structure and its set-theoretic surrogate are isomorphic (of course, that's an intuitive isomorphism, not a set, since I began with the intuitive natural numbers). So the natural numbers are far better matched in set theory than the universe $V$ is. – Andreas Blass Aug 5 '13 at 3:11
@NickR I'm not sure what you mean by "represented in set-theoretic terms". The class of all sets is not in the universe described by ZFC. (Large cardinals, on the other hand, may well be there; if they exist, they are sets.) – Andreas Blass Aug 5 '13 at 3:13