A set-theoretical formula equivalent to the axiom of choice.
Bolesław Sobociński · Notre Dame Journal of Formal Logic · 1962
It is obvious that the following set-theoretical formula:For any cardinal numbers m and n which are not finite, if $(m) and $(n) are the least Hartogs* alephs with respect to m and n respectively, and such that $(m) = $(n), then there is no cardinal )p such that m < }o< n.is a simple consequence of the theorem: 21.For any cardinal numbers m and n which are not finite, if $(w) and $(rt) are the least Hartogs' alephs with respect to m and rt respectively, and such that $(m) = $ (n), ί&eπ m = n.which, as it is proved in [3], p. 230, is inferentially equivalent to the axiom of choice.Although at first glance it appears that formula SI is weaker than 21, in fact, as I shall show in this note, the former formula implies the axiom of choice, and, therefore, it is inferentially equivalent to 21. For, a proof is given here that the following theorem: A. For any cardinal number m which is not finite, if$(w) is the least Hartogs' aleph with respect to m, then there is no cardinal )p such that tf (m)< £