A. R. D. Mathias

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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)
SAUNDERS 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, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, TCo, of Transitive Containment, we shall(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 〈Aλ | λ a limit ordinal 〉 where for each λ, Aλ ⊆ 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 Jω. Much of the development is(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)
Bourbaki suggest that their definition of the number 1 runs to some tens of thousands of symbols. We show that that is a considerable under-estimate, the true number of symbols being that in the title, not counting 1,179,618,517,981 disambiguatory links. 1: Introduction Bourbaki, the self-perpetuating French group of mathematicians, are ill at ease with(More)
3 0 Introduction 3 Part I 1 Formulation of the various systems 7 2 Theorems of various systems 13 On ReS and finite sets 13 Proof that HF models ZF – Infinity 16 On GJ and the class of rudimentary functions 20 Companions and the Gandy–Jensen Lemma 21 A single generating function for rud(u) 22 Other remarks on GJ 23 On fReR 25 On ReRI 26 On KP 27 3 Remarks(More)
This paper, a contribution to “micro set theory”, is the study promised by the first author in [M4], but improved and extended by work of the second. We use the rudimentarily recursive (set theoretic) functions and the slightly larger collection of gentle functions to develop the theory of provident sets, which are transitive models of PROVI, a very weak(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.