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?
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29We can write down all mathematical theorems with ballpoint pens on paper, too. Does that mean all mathematics is ink in disguise? – mjqxxxx Dec 15 '20 at 22:51
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13@mjqxxxx No, it's chalk on a blackboard. – badjohn Dec 15 '20 at 23:02
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No, but some people around here seem to think that it's all the axiom of choice in disguise. – Dec 16 '20 at 08:11
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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. – Dec 16 '20 at 08:26
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slightly related: What does “most of mathematics” mean? – uhoh Dec 16 '20 at 09:53
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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.
J.G.
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2"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." – Arthur Dec 15 '20 at 22:57
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@Arthur That is an excellent book. I have to recommend it for its proof that $2\in\Bbb P$ alone. – J.G. Dec 15 '20 at 23:01
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1Yes, 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. – Arthur Dec 15 '20 at 23:06
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1@Arthur I adore proofs which really ought to be circular but aren't - that book is one of my favorites. – Noah Schweber Dec 15 '20 at 23:14
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I think the proof that $2$ is prime is over the top: one only has to prove that $\Bbb{Z}/2\Bbb{Z}$ is an integral domain to see that $2$ is prime $\ddot{\smile}$. – Rob Arthan Dec 16 '20 at 21:40
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My example I tend to use under the heading of "junk theorems": suppose some student came to you as a teacher and asked a question starting with "suppose $G$ and $H$ are groups and $H \in G$". Would your first instinct be to furrow your brow, and ask something like: "probably you meant to say $H \le G$ i.e. $H$ is a subgroup of $G$, instead"? (That kind of example tends to convince me that day-to-day mathematics is thought about in terms of some kind of type theory, even if the mathematicians themselves think they're using ZFC.) – Daniel Schepler Dec 22 '20 at 18:17
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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.
Noah Schweber
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1@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 $\mathsf{V=L}$. – Noah Schweber Dec 23 '20 at 04:29
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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.
Rob Arthan
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