View source for Comparability of cardinals

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The proposition that any two [[cardinal number]]s $\mathfrak{a}, \mathfrak{b}$ are comparable, so that one of  $\mathfrak{a} < \mathfrak{b}$,  $\mathfrak{a} = \mathfrak{b}$,  $\mathfrak{a} > \mathfrak{b}$ holds.  This asserts that of any two sets, one may be put into one-to-one correspondence with a subset of the other.  Comparability follows from the [[Well-ordering theorem]], which implies that infinite cardinal numbers are all [[aleph]]s: conversely, the comparability of sets in general implies the well-ordering theorem.  Hence this is an equivalent of the [[Axiom of choice]].

====References====
* Abraham A. Fraenkel, "Abstract set theory", North-Holland (1953)  {{ZBL|0050.04903}}