From Encyclopedia of Mathematics - Reading time: 1 min
2020 Mathematics Subject Classification: Primary: 03E25 [MSN][ZBL]
The proposition that any two cardinal numbers are comparable, so that one of , , 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[edit]
- Abraham A. Fraenkel, "Abstract set theory", North-Holland (1953) Zbl 0050.04903