I'm doing the following exercise from Just/Weese:
Show in ZF that (WO) implies (IC) and that (IC) implies (SC).
where
(WO) Every set can be well-ordered.
(IC) For any two sets $X,Y$ either there is an injection $X \hookrightarrow Y$ or $Y \hookrightarrow X$.
(SC) For any two sets $X,Y$ either there is an surjection $X \twoheadrightarrow Y$ or $Y \twoheadrightarrow X$.
(WO) $\rightarrow$ (IC): Let $X,Y$ be two sets. Then by (WO) they can be well-ordered. Therefore each is in bijection with an ordinal $\alpha$ (and $\beta$, respectively):
Claim: Every well-ordered set is isomorphic to an ordinal.
Proof: Let $\langle, X,W \rangle$ be a well-ordered set. Let $\alpha$ be an ordinal with $|\alpha| \ge |X|$. Define an injective map $f: X \hookrightarrow \alpha$ as follows:
(i) Let $x_0$ be the $W$-minimal element. Then $x_0 \mapsto \varnothing$.
(ii) Assume $f$ has been defined for $x \in I_W (x')$. Define $x' \mapsto \sup^+ f(I_W (x'))$.
Let $\tilde{f} = f: X \to \mathrm{im}f$. Then $\tilde{f}$ is a bijection and $\mathrm{im}f$ is an initial segment of an ordinal hence also an ordinal.$\Box$
Either $\alpha \in \beta$ or $\beta \in \alpha$. Hence either $X \hookrightarrow Y$ or $Y \hookrightarrow X$.
(IC) $\rightarrow$ (SC): Let $X,Y$ be sets. Then either $X \hookrightarrow Y$ or $Y \hookrightarrow X$. Given $X \hookrightarrow Y$ it is easy to construct a surjection $Y \twoheadrightarrow X$, similarly for $Y \hookrightarrow X$.
Can you tell me if these proofs are correct? Thanks.
I also wanted to prove (IC) $\rightarrow$ (WO), but I'm stuck. I thought of something like if $X$ is a set and $\alpha$ is an ordinal then either $X \hookrightarrow \alpha$ or $\alpha \hookrightarrow X$ but the latter case seems to be a dead end.