Does ZFC without Foundation prove that given any proper class A, every set can be injected into A ?

I have a proof in ZFC, using AC and the axiom of foundation, that given any proper class A, every set can be injected into A. I wonder if we could have a proof of this that does not use Foundation. Gérard Lang

• I think that you have another account, no? You can ask on the meta site to have them merged. – Asaf Karagila Feb 23 '13 at 14:39
• Interesting. Would it be correct to infer that the overall strategy is to prove, presumably by transfinite induction, that it is possible to inject any $V_\alpha$? – Adam Epstein Feb 23 '13 at 15:44
• Upvote for a compelling statement that requires Replacement. – Adam Epstein Feb 23 '13 at 15:48
• Adam, in ZFC (proper!) it's easy to prove. Let $x$ be a set, well order it. Since $A$ is a proper class for every $\alpha$ there is $\beta>\alpha$ such that $V_\beta\cap A$ has more elements than $V_\alpha\cap A$. Let $\alpha_i$, $i<|x|$ be a strictly increasing sequence of the least ordinals where information on $A$ is added. For every $i<|x|$ map $x_i\in x$ to some element in $V_{\alpha_i}\cap A\setminus A\cap\bigcup_{j<i}V_{\alpha_j}$. – Asaf Karagila Feb 23 '13 at 15:52
• Also, it should be remarked that the use of choice is also essential. It is consistent that there is a proper class without a countably infinite subset. – Asaf Karagila Feb 23 '13 at 15:53

• Dear Gerard, If you have a proper class of sets of the form $\lbrace x\rbrace = x$ then I believe that you can treat them as atoms and repeat all the constructions and have this. – Asaf Karagila Feb 23 '13 at 18:03