The following is an exercise in Just/Weese:
Show in ZF that the following are equivalent for every set $A$:
(a) There is no injection $f: \omega \to A$
(b) Every injection $f: A \to A$ is a surjection
Can you tell me if my proof is correct? Thank you!
(b) $\rightarrow $ (a): Assume $\lnot$ (a) and let $f: \omega \hookrightarrow A$ be an injection. Then $A$ is infinite hence (by corollary 13 on page 49) there exists a map $g: A \to A$ that is injective but not surjective.
(a) $\rightarrow $ (b): Assume $\lnot$ (b) and let $f: A \to A$ be injective but not surjective. Let $a_0 \in f(A)^c$. Then the following map is an injection: define $g: \omega \to A$ as $g(\varnothing) = a_0$ and $g(n) = f(g(n-1))$.
To see that $g$ is injective assume $g(n) = g(m)$ for $n > m$. Then $g(n-1) = g(m-1)$ and so on, until $g(n-m) = g(\varnothing) = a_0$. But the empty set is the only element mapping to $a_0$ hence $n-m = \varnothing$ and hence $n = m$.