Comparability of cardinals

2020 Mathematics Subject Classification: Primary: 03E25 [MSN][ZBL]

The proposition that any two cardinal numbers $\mathfrak{a}, \mathfrak{b}$ 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 alephs: 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