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I'm trying to prove the following:

If it holds that if for any two sets $A$ and $B$, $A$ can be injected into $B$ or $B$ can be injected into $A$, then every set can be well-ordered (axiom of choice variant).

I intend to use Hartogs ordinal, but I am not very sure... my idea is that: Assuming that $A$ and $B$ have different cardinal number, and that $A$ can be injected into $B$, then $|A|<|B|$. Considering the Hartogs ordinals $H(A)$ and $H(B)$ of $A$ and $B$, I think that I can assume that $|H(A)|<|H(B)|$, then it should exist an injection from $H(A)$ to $B$, otherwise, $H(A)$ would be the Hartogs ordinal of $B$.

I'm not sure if this reasoning is correct how to continue it, I'm just beginning to study the axiom of choice...

Any ideas? Thanks!

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By the way, Hartogs is the correct way to write that name, it's not Hartog. – Asaf Karagila Mar 17 '13 at 13:53
You can find different answers here… – Camilo Arosemena Mar 17 '13 at 16:22
up vote 4 down vote accepted

Hint: Compare $A$ with its Hartogs. Conclude that $A$ can be injected into an ordinal and can therefore be well-ordered.

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