ABSTRACT
Others have predicted that set theory will split into incompatible sub theories, only the common part of which could serve as a working foun dation for mathematics ([20], p. 115) or even claim a central place in mathematics ([15], p. 94). As one writer put it: “ I guess that in the future we shall say as naturally ‘Let us take a set theory S9 as we now take a group G or a field F ” ([8], p. 105). For convenient reference, I will call views such as these “alarmist.” Not everyone agrees with the alarmists that the good old days when we
could speak of set theory (rather than theories), are gone. Godel ([5]) and a number of others ([9], [17]) still maintain that set theory is the theory of a definite mathematical structure. Kreisel has been particularly active in argu ing that ‘set’ is neither vague nor ambiguous, that set theory has a unique intended interpretation and that under that interpretation, the CH is either true or false, although we do not know which ([9, 10, 11, 12, 13, 14]). I am inclined to agree with Kreisel on all these points, but not with the
arguments and claims he makes in their support. Specifically, he asserts that the continuum hypothesis “w decided by the second order axioms of Zermelo” ([10], p. 99). Kreisel repeats this assertion in various articles and regards the “ second order decidability of CH” as “the main theme” of his article “ Informal Rigour and Completeness Proofs” ([9], p. 152).
