The Axiom of Choice, and the statement that whenever there is a surjection $A \to B$ there is an injection $B \to A$

Asked Oct 24 '15 at 11:05

Active Oct 24 '15 at 11:16

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Is there a model of $\mathsf{ZF}$ in which the Axiom of Choice fails but the following statement holds for arbitrary sets $A, B$:

there is an injection $A \to B$ if there is a surjection $B \to A$?

edited Oct 24 '15 at 11:16

user642796

asked Oct 24 '15 at 11:05

wjea

Nobody knows yet. – Asaf Karagila Oct 24 '15 at 11:07
I have written many answers on that topic. – Asaf Karagila Oct 24 '15 at 11:08
Could you give me a link? – wjea Oct 24 '15 at 11:09
http://math.stackexchange.com/search?q=user%3A622+%22partition+principle%22 – Asaf Karagila Oct 24 '15 at 11:10

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