# What can category theory encode that set theory could not encode, if anything? (Fwd from Quora) [duplicate]

Interesting reply to this question at Quora from Alex Sadovsky, Ph.D. Mathematics & Biomechanics, University of California, Irvine

Nothing. All of modern mathematics is described in the language of naive set theory. Every area of mathematics, whether category theory, representation theory, functional analysis, or algebraic geometry, is just a specialization. It does not introduce anything that could not be described by set theory.

Category theory specializes in Morphisms.

• I mean, categories can be (and are usually) described in terms of set theory, so clearly anything that can be encoded with categories can also be encoded with sets. Apr 13, 2020 at 15:37
• “I recall that, at the 1963 meeting devoted to Logic, Methodology and Philosophy of Science in Jerusalem, Bill Lawvere proposed basing mathematics on categories rather than sets. Alfred Tarski, who was in the audience, objected: what is a category if not a set of objects and a set of arrows? Lawvere replied: set theory deals with the binary relation of membership, category theory with the ternary relation of composition. Apparently, Tarski was satisfied with the answer.” Apr 13, 2020 at 16:00
• Can you link to the Quora question? Apr 13, 2020 at 16:25
• @NoahSchweber Link added. Apr 13, 2020 at 17:17
• Please don't post "fwd from Quora" stuff here. If you have a question, ask the question. This is not a site to "get comments" or "encourage discussion". It's a Q&A site. Apr 14, 2020 at 13:23

• Although that said, I take huge issue with the "almost equiconsistent" bit - dropping replacement results in a gigantic loss of consistency strength. The right thing to compare ETCS to if I recall correctly is Z (or ZC). To see the drop in consistency strength, keep in mind that ZFC proves that $V_\lambda\models ZC$ for every infinite limit cardinal $\lambda$. This doesn't affect the broad point of the answer of course, since Z itself is a ZFC-style set theory, but I think it's worth noting. Apr 13, 2020 at 16:18
• @spaceisdarkgreen: You can construct a structure that is isomorphic to $(V_\omega,\in)$ in ZC. You just can't get a structure where the relation is actually $\in$ itself. So from the perspective of "ordinary" mathematics where you only care about studying structures up to isomorphism, you can do pretty much everything you might care about as long as you don't need any sets whose cardinality is too big (basically, anything that involves strictly increasing an infinite cardinality infinitely many times). Apr 13, 2020 at 19:15
• @spaceisdarkgreen: Well, for instance, you can define a relation $R$ on $\omega$ such that $(\omega,R)\cong (V_\omega,\in)$, by encoding the combinatorics of hereditarily finite sets with numbers in any of the usual ways. Apr 13, 2020 at 19:21