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In The prospects of unlimited category theory: doing what remains to be done, 2014 (The Review of Symbolic Logic, 8 (2015) pp 306-327, link), Ernst discusses Feferman's program, described in Foundations of unlimited category theory: what remains to be done (The Review of Symbolic Logic, 6 (2013) pp 6-15, link), for "unlimited category theory", characterized by three desirable axioms:

  • There exists a category containing all objects of a given type (set, group, topological space, etc). It should literally include all such objects with no size limitations.
  • For any two categories $A$, $B$, there should exist a functor category $B^A$.
  • It should enable all standard constructions such as $\mathbb{N}$ and sums and products, etc.

Ernst shows that these axioms are inconsistent, by proving a version of Cantor's theorem for the category of reflexive graphs, thereby demonstrating a version of Cantor's paradox, showing that desired axioms are inconsistent.

He concludes that the search for such foundations is misguided, and so are objections to a ZFC-based set theoretic foundation for category theory.

To me this seems analogous to the situation with formal set theory with unrestricted comprehension in the days of Russell's paradox. Or the situation Girard's paradox for Martin-Löf type theory. Those formalisms were able to be saved. Is there any hope of salvaging Feferman's unlimited category theory?

What is the state of the art in category-theoretic foundations for category theory in 2018?

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    $\begingroup$ Paraconsistency is one solution. $\endgroup$ – user40276 Jun 7 '18 at 21:05
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    $\begingroup$ Are the titles in all caps really necessary? $\endgroup$ – David Roberts Jun 8 '18 at 2:33
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    $\begingroup$ The formalisms were "able to be saved" by removing the problematic "object of all objects". In Feferman's case that means dropping his first desideratum, which makes it no longer "unlimited". (This is unrelated to the question of category-theoretic foundations for category theory.) $\endgroup$ – Mike Shulman Jun 8 '18 at 13:56
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    $\begingroup$ @user40276 Excluding EFQ doesn't save you, because of Curry's paradox. Assuming you have the axiom of separation, if there is a set of all sets then you can form the set $C = \{ x \mid (x\in x) \to P \}$ for a arbitrary proposition $P$. Now reproduce Russell's paradox with $P$ in place of $\bot$. If $C\in C$, then by definition of $C$, $(C\in C)\to P$, whence $P$. Therefore, $(C\in C) \to P$; hence $C\in C$ by definition of $C$, and therefore $P$. So every statement is provable, making for a rather uninteresting theory. $\endgroup$ – Mike Shulman Jun 12 '18 at 23:24
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    $\begingroup$ You can also phrase this more categorically: if every limit-preserving functor between complete categories has a left adjoint, then every endofunctor of Set has an initial algebra and therefore (by Lambek's lemma) a fixed point. In particular, this is true for the covariant powerset functor $\Omega^{(-)}$, so there is a set $X$ such that $r : A\cong \Omega^A$. Now let $C = r^{-1}(\{x\in A \mid (x\in r(x))\to P\})$ for an arbitrary statement $P$ and argue as before. $\endgroup$ – Mike Shulman Jun 12 '18 at 23:28

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