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I came across the following question in a discussion about set theory and model theory with a friend of mine. It is purely intrinsic.

In some text books on set theory, e.g. the german text book by Ebbinghaus, one can read the following line: First-order set theory is sufficient for most of ordinary mathematics.

Question: What would be an example of something we cannot formalize in first-order set theory?

If we assume a set-theoretical system of axioms to be consistent, then of course there is always a theorem that cannot be proven. But my question is ment in a different sense. Namely, are there mathematical concepts that cannot be described via sets?

Also it is clear to me, that the formalization of all concepts relies heavily on the collection of axioms. E.g. we cannot define what a function is, if we are lacking certain $\sf ZFC$-axioms. But if we now assume, that we can add to $\sf ZFC$ as many axioms as we want, are there still concepts that cannot be formalized?

Thank you in advance for your help!

mrtaurho
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Daniel W.
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    What about proper classes; I wouldn't say that something that isn't a set is a solid description, but that's up for discussion. – mrtaurho Jun 10 '20 at 12:19
  • Okay, then instead of ZFC, let's use NBG, or any set-theoretical axioms you want to. The question in some sense should purely rely on the language and the possibilities of sets. – Daniel W. Jun 10 '20 at 12:21
  • Maybe I got distracted by you referencing $\sf ZFC$ :D Anyway, interesting question and I'll wait for others to answer as I'm (sadly) not able to! – mrtaurho Jun 10 '20 at 12:24
  • I already saw that one, but I am not sure whether it answers my question in a sufficient way. Are there no concrete examples of objects we can give in this context? – Daniel W. Jun 10 '20 at 12:36
  • An assertion about "formulas" (logical expressions) involving sets can usually not be written in first-order terms. For this reason, the "axiom of replacement" is actually the axiom schema for replacement. – Kapil Jun 10 '20 at 12:41
  • But in principle you can say what a formula is and you can say what a schema of formulas is, namely a set of formulas. Of course, for this you have to go to some "meta set theory", but still set theory. – Daniel W. Jun 10 '20 at 12:52
  • I don't think there are mathematical concepts that set theory (given sufficient axioms) would not capture. – Zuhair Aug 23 '21 at 10:18

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Skolem's paradox...

How can there be a countable model of set theory if even the real numbers are uncountable?

Because that "model" is only a model of the first-order theory of sets.

GEdgar
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    And how is this related to my question? Skolem's Paradox is only a question of the model you are living in. Or am I missing something? – Daniel W. Jun 10 '20 at 13:43
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    Skolem's paradox can very much be formalised in set theory. It is a paradox about set theory, yes, but set theory can formalise set theory. – Asaf Karagila Jun 10 '20 at 14:27