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S AUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasizes, one that is adequate for much of(More)
If one looks at the history of mathematics, one sees periods of bursting creativity, when new ideas are being developed in a competitive and therefore very hasty spirit; and periods when people find that the ideas so recently in vogue are inexact, incoherent, possibly inconsistent; in such periods there is an urge to consolidate past achievements. I said "(More)
Let 91 be a ~-complete ultrafilter on the measurable cardinal ~. Scott [ 1 3 ] proved V ¢ L by using 91 to take the ultrapower of V. Gaifman [ 2] considered iterated ultrapowers of V by cg to conclude even stronger results; for example, that L n ~(6o) is count-able. In this paper we discuss some new applications of iterated ultra-powers. In § § 1-4, we(More)
The extensive theory that exists on ~ca, the set of uitrafilters ov-.r the integers, suggests an analogous study of the family of g-complete ultrafih~rs over a measurable cardinal g > w. This paper is devoted to such a study, with emphasis on those aspects which make the un-countable case interesting and distinctive. Section 1 is a preliminary section,(More)
Working in Z +KP , we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula Φ(λ, a) such that for any sequence A λ | λ a limit ordinal where for each λ, A λ ⊆ λ 2, there is a supertransitive inner model of Zermelo(More)
Using the theory of rudimentary recursion and provident sets developed in a previous paper, we give a treatment of set forcing appropriate for working over models of a theory PROVI which may plausibly claim to be the weakest set theory supporting a smooth theory of set forcing, and of which the minimal model is Jensen's J ω. Much of the development is(More)
Bourbaki suggest that their definition of the number 1 runs to some tens of thousands of symbols. We show that that is a considerable underestimate , the true number of symbols being that in the title, not counting 1,179,618,517,981 disambiguatory links. Bourbaki, the self-perpetuating French group of mathematicians, are ill at ease with logic and(More)
We give an example of an iteration with recursive data which stabilises exactly at the first non-recursive ordinal. We characterise the points in the final set as those attacked by recurrent points, and use that characterisation to show that recurrent points must exist for any iteration with recursive data which does not stabilise at a recursive ordinal.(More)
A Examples are discussed of natural statements about irrational numbers that are equivalent, provably in ZFC, to strong set-theoretical hypotheses, and of apparently classical statements provable in ZFC of which the only known proofs use strong set-theoretical concepts. Opponents of a full-blooded set-theoretic account of the foundations of(More)