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Post Undeleted by Andreas Blass
2 completely replaced an answer that duplicated a previous one

The Cantor-Schroeder-Bernstein theorem (that if each of two sets admits an one-to-one map into the other then there is a bijection between them) is often

Tarski proved using the axiom of choice. That proof uses the well-ordering theorem to assign to every that, for any set $X$, as its cardinal number, A$, the smallest ordinal number in one-to-one correspondence with set$X$. The hypothesis W(A)$ of the C-S-B theorem easily gives that the cardinal numbers well-orderable subsets of the two sets are each less $A$ has strictly larger cardinality than or equal to the other$A$. This is trivial with AC, as then $W(A)$ is the whole power set of $A$ and therefore equalthus Cantor's theorem applies. But there is another Tarski gave a proof that entirely avoids the axiom of choiceAC.

(Tangential remark: Using

I don't have my copy of Howard and Rubin's "Consequences of the well-ordering theorem to define cardinal numbers as above, one also gets an easy proof that, Axiom of any two setsChoice" handy, but if I did then I could probably find lots of examples by looking at least one admits a one-to-one map into the othervarious forms numbered 0A, 0B, etc. This conclusion is actually equivalent to the axiom of choice in the presence I believe all of these are provable without AC (hence the other ZF axioms.number 0) but there was once a reason to suspect AC was needed.

Post Deleted by Andreas Blass
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