On one of Specker's theorems

Abstract

The Specker theorem is considered, stating that if for some cardinal m there is no strictly intermediate cardinal between m and 2m (briefly CH(m)), and also between 2m and 22m (briefly CH(2m)), then the cardinal 2m is an aleph (briefly WO(2m)). Rejecting the second condition CH(2m) in Specker's theorem, a different condition of local bounded selection of elements from the family of bijections of the set X, |X|=m, to sets X⊔[0,α), 0<α<ℵ(m) (briefly AC2m(ℵ(m))) is specified, which is not only a sufficient condition for 2m=ℵ(m), which strengthens Specker's theorem, but also a necessary condition for it. More precisely, if m is an infinite cardinal, then the formula 2m=ℵ(m) holds if and only if the following formula CH(m)∧AC2m(ℵ(m)) is true. Here ℵ(m) is a Hartogs number. The following generalized Specker theorems are also proved:(CH(m+ℵ(m))∧CH(2m))⇒WO(2m),(CH(m)∧CH(m+ℵ)∧ℵ≥ℵ(m))⇒WO(2m) and(CH(m+ℵ)∧CH(2m)∧ℵ(m)<ℵ≤2m)⇒WO(2m). The following nontrivial theorem is proved: Let m be a cardinal such that the formula CH(2m) is true and Hartogs number ℵ(2m) is a regular cardinal. Then folmula WO(m) is true if and only if m2=m and for every alephs ℵ,ℵ′ such that 2m<2ℵ and 2ℵ′<2m the following inequalities m<ℵ and ℵ′<m are valid, respectively. An unsolved problem is still being discussed: is the formula CH(m)⇒WO(m) true? Solving this problem boils down to solving the question of the consistency of the formula CH(m)∧¬WO(m) with the axioms of Zermelo-Fraenkel set theory without the axioms of choice and regularity (briefly ZF). The problem of consistency and independence of formulas CH(m)∧¬WO(m) and CH(m)∧¬WO(2m) with the axioms of set theory ZF is also discussed. Their independence from these axioms ZF is proved. The consistency of the last formula is known and the question of the consistency of the first formula with axioms ZF remains unresolved yet.