# Is all of mathematics set theory in disguise?

Is all of mathematics, just set theory in disguise? We know the ZFC axioms allow us to encode mathematical objects as sets of some kind. So, even theorems about, say, Hilbert spaces can be written down, tediously, as formulas of ZFC. So, does this mean all of mathematics is set theory in disguise?

• We can write down all mathematical theorems with ballpoint pens on paper, too. Does that mean all mathematics is ink in disguise? Dec 15, 2020 at 22:51
• @mjqxxxx No, it's chalk on a blackboard. Dec 15, 2020 at 23:02
• No, but some people around here seem to think that it's all the axiom of choice in disguise.
– user403337
Dec 16, 2020 at 8:11
• And while AC is equivalent to the maximum modulus principle, Zorn's lemma, the well-ordering principle, the Tychonoff theorem, the ability to choose a basis for a vector space, and God knows what else, I think that that's going too far.
– user403337
Dec 16, 2020 at 8:26
• slightly related: What does “most of mathematics” mean?
– uhoh
Dec 16, 2020 at 9:53

## 3 Answers

It depends what you mean by "is."

There is certainly a sense in which we can "embed" all of mathematics inside $$\mathsf{ZFC}$$. Consequently, if we imagine a person who for whatever reason is only ever willing to talk about actual formal proofs from $$\mathsf{ZFC}$$, they would still be able to prove not-explicitly-set-theoretic theorems by proving the appropriate set-theoretic translation.

However, $$\mathsf{ZFC}$$ - and more generally the framework of set theory - is not unique in this sense. Personally I find it the most natural framework, but this is absolutely a subjective position and there are good arguments against it; the one I find most compelling is the "non-structural" nature of $$\mathsf{ZFC}$$, mentioned in J.G.'s answer, namely that when we implement a piece of non-set-theoretic mathematics in $$\mathsf{ZFC}$$ we wind up with "junk theorems" in addition to the theorems which are translations of meaningful results in the original context.

So is mathematics set theory? Well, it can be if you want it to - but that says more about your choice of style than about mathematics.

• Why is ZFC formatted using \mathsf{}? Is it a convention? Dec 16, 2020 at 13:54
• @AniruddhaDeb Yeah, I don't know precisely where I learned it from but I've stuck with it, it looks nice. That said, it's worth noting that there are a couple situations where using a specific font for theories is convenient. For example, I find it useful to distinguish the meta-statement "$V=L$" from the particular first-order sentence $\mathsf{V=L}$. Dec 23, 2020 at 4:29

No.

Say you've identified a mathematical theory $$T$$ with a model in $$\mathsf{ZFC}$$. It has multiple models in $$\mathsf{ZFC}$$, each with different this-object-is-that-set identifications. Not only are these identifications different by model, they also have at-the-set-level consequences that don't mean anything about $$T$$ itself. For example, some but not all models of $$\Bbb N$$ in $$\mathsf{ZFC}$$ satisfy $$m\le n\implies m\subseteq n$$, where the first statement is about naturals but the second is about sets with which they're identified when specifying a model. It would be silly to write $$2\subseteq 3$$, or $$2\in 3$$ (these are examples of what @NoahSchweber's answer calls junk theorems), because these identifications don't tell us what $$T$$'s objects "really are". The only reason models are interesting is that, as long as you can find one, $$\mathsf{ZFC}$$ (or whatever baseline we picked) implies $$T$$ is consistent.

You could just as easily use something else as a "base" for mathematics, e.g. category theory.

• "e.g. category theory" See, for instance, Mathematics Made Difficult. It features gems like "As an axiom on which to base the positive numbers and the integers, which have in the past produced much harmless amusement and are still widely accepted as useful by most mathematicians, some such proposition as the following is sometimes considered as being pleasant, elegant, or at least handy: AXIOM: [co]Equalisers exist in the category of categories." Dec 15, 2020 at 22:57
• @Arthur That is an excellent book. I have to recommend it for its proof that $2\in\Bbb P$ alone.
– J.G.
Dec 15, 2020 at 23:01
• Yes, because the best way to do that is obviously to prove that $\Bbb Z/2\Bbb Z$ is a field. It's so ridiculously backwards it just feels like there must be some circularity somewhere in the argument. Dec 15, 2020 at 23:06
• @Arthur I adore proofs which really ought to be circular but aren't - that book is one of my favorites. Dec 15, 2020 at 23:14
• @NoahSchweber They would be a good soft question topic.
– J.G.
Dec 16, 2020 at 9:35

I would say that it's the other way round: most mathematical topics can be disguised in set theory, but their true nature is only revealed when the disguise is cast off. For a simple example, the set theoretic view of the permutation groups $$S_n$$ as a set of functions each function being represented as a set of pairs doesn't help much in elementary group theory.