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Is the class of cardinals totally ordered?

Intuitively, it seems like for any sets $A,B$ either $\lvert A\rvert\leq \lvert B\rvert$ or $\lvert B \rvert \leq \lvert A\rvert$. How can I prove this?

Using the definition of cardinality, the problem reduces to proving that for all sets $A,B$, there is either an injection from $A$ to $B$ or from $B$ to $A$. However, I don't see how to proceed from there. Is AC necessary?

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Yes, AC is necessary. –  Brian M. Scott Aug 10 '12 at 7:17
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marked as duplicate by Arthur Fischer, t.b., Chris Eagle, William, Zhen Lin Aug 10 '12 at 8:05

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1 Answer

Yes, the axiom of choice is necessary. Without it you can have an infinite, Dedekind-finite set. If $A$ is such a set, there is no injection of $\omega$ into $A$ and no injection of $A$ into $\omega$.

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