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As has been shown on this site before, it is possible to prove the Cantor-Bernstein-Schröder theorem without the axiom of infinity or the axiom of substitution (most simply by using an argument based on the Knaster-Tarski lemma, but there are other ways that use naturals without assuming they form a set). We have also seen, in "Does Cantor-Bernstein hold for classes?" that the theorem can be proven for injections between classes in, e.g., NBG. However, the question has not arisen whether it is possible to prove it for classes without using the axiom of infinity (and, I suspect, the axiom of substitution). The "sequence of sets" approach doesn't work because there'a no such thing as a sequence of classes. It's possible to flip that approach around, I believe, by instead considering a class of sequences, but that requires the axiom of infinity and the axiom of substitution. I don't see a way to make the Knaster-Tarski approach work, because classes don't form anything that looks much like a complete lattice. Any thoughts?

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What is the axiom of substitution? Is it different from Replacement? – Noah Schweber Apr 3 '13 at 22:13
No, it's the same. – dfeuer Apr 3 '13 at 23:02
Anything go wrong with this proof:… – François G. Dorais Apr 4 '13 at 1:41
François, I thought something did, but looking at it more carefully I think you're right. – dfeuer Apr 4 '13 at 2:40
I think I got confused because the alternate definition of $C$ in that proof definitely does not work with classes, but I think the first formulation should. – dfeuer Apr 4 '13 at 2:44

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