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In material set theories (like ZFC), one can prove that there is no set of all sets. Can one prove a similar statement in ETCS? This exact statement "there is no set x such that y in x for every set y" is not expressible in ETCS because ETCS is a structural set theory. But however is there a way to nevertheless talk about size issues in ETCS?

Also, in ETCC, which axiomatizes the category of categories, can one there prove/formulate a statement like "There is no category that contains every category"?

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  • $\begingroup$ One should mention algebraic set theory as well, which is an axiomatisation of a category of classes. $\endgroup$
    – David Roberts ♦
    Commented Mar 13, 2017 at 3:51
  • $\begingroup$ Here is a related statement: There is no object $X$ (of our topos of sets) such that every object $Y$ admits a monomorphism $Y \to X$. $\endgroup$
    – HeinrichD
    Commented Mar 17, 2017 at 13:34
  • $\begingroup$ I'm curious to know what you think "there is no set of all sets" means. You seem to suggest that it is a good thing that ZFC proves it, so presumably you think it is true, but what does it really mean at all? If it just means it is not true in any model of ZFC, then it is actually irrelevant whether other foundational systems prove it. If not, what more do you think it means? $\endgroup$
    – user21820
    Commented Dec 27, 2019 at 15:54

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We can talk about a family of sets $Y_x$ parameterized by the elements $x \in X$ of another set using a bundle of sets, namely a map

$$Y = \bigsqcup_{x \in X} Y_x \mapsto X$$

whose fibers are the family of interest. This is, for example, how ETCS talks about the axiom of choice. So we can phrase the nonexistence of a "bundle of all sets" as the following claim.

Claim: There is no map $f : Y \to X$ such that, for every set $S$, there exists $x \in X$ such that $S \cong f^{-1}(x)$.

(Note that $x \in X$ is the same as saying $x : 1 \to X$ and $f^{-1}(x)$ is the pullback of the diagram $1 \xrightarrow{x} X \xleftarrow{f} Y$; this really is an entirely categorical statement.)

What I want to say from here is that we should pick $S = 2^Y$ and then appeal to Cantor's theorem, or more categorically the Lawvere fixed point theorem. But what we need is a surjection $Y \to 2^Y$ and what we get is an injection $2^Y \to Y$, and I'm not sure how much of ETCS is necessary to make everything work out.

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    $\begingroup$ In any topos, the subobject classifier is internally injective, so that any injection $A\rightarrowtail B$ yields a surjection $\Omega^B \twoheadrightarrow\Omega^A$. Thus, an injection $\Omega^Y \rightarrowtail Y$ gives a surjection $\Omega^Y \to \Omega^{\Omega^Y}$, allowing us to apply Cantor's theorem. $\endgroup$
    – Mike Shulman
    Commented Mar 13, 2017 at 12:12

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