On a consistency theorem connected with the generalized continuum problem

Abstract

It is known that G~3DEL [l] has proved the consistency of the generalized continuum hypothesis (c , ) (2s"= ~,;1) with the axiom system ~'*.~ Presumably the generalized continuum hypothesis is independent of the axiom system X*, however, the consistency of the negation of this hypothesis with v . has not yet been proved. Moreover, presumably each of the particular cases 2 s0 = N1, 2,~1=.N2,... . . . , 2 ~ ~ No,;~ . . . . . 2 ~ N;~;~, . . . is independent of the axiom system 27. However, for a given ordinal Z the question of independence of the statement 2 s ~ Nx,l has a precise meaning only in the case when the ordinal number s can be defined by means of the notions of the set theory, i. e. if it is a particular class which is at the same time an ordinal number as, for example, 0, 1 or o) (or o)~1 or o)~). Let A, N, . . . denote guch particular ordinal numbers. Now, since no theorem of the set theory is known which would allow to infer from the assumption 2 ' ~ = N for certain ordinals ,u the equality /.~+1 2 '~'4~ NA;~ for a A different from these ~, we can formulate the conjecture that the equality 2~A= NA;j cannot be proved for any A from the axioms of 27 even if we assume that 2 st~ -N~; I is fulfilled for every tr different from zl. It is obvious that until the independence of the generalized continuum hypothesis is proved, the question whether an equality of the form 2 s A = ---~ Na;~ can be proved from the axioms of X* and certain additional assumptions may be investigated only if we assume that it cannot be proved from the axioms of X* alone. So we hope that some interest may be attached to theorems which