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Summary of the field of research of my mathematical publications

Mathematical logic has come a long way since the work of Frege, the discovery by Russell of a contradiction in Frege's system, and the responses of Zermelo, Russell and Whitehead to that discovery. It is noticeable that many of the subsequent proposals for a single foundation of mathematics emphasize one aspect of mathematics at the expense of another, and for my part, I hold that, for reasons going back to the ancient Greek division between arithmetic and geometry, mathematics requires a dual foundation, a balance, which has yet to be achieved, between the insights of set theory and of category theory.

A) A great advance in set theory occurred in 1935-8 with Gödel's creation of the notion of constructibility, the main technical tool underlying his relative consistency proofs for the Axiom of Choice (AC) and the Generalised Continuum Hypothesis (GCH). A second great advance occurred when I was an undergraduate in the early 1960's, namely Cohen's creation of forcing, the main technical tool underlying his independence proofs for AC and GCH. This advance attracted me to set theory; and the papers in List A concern the development of a then new instance of Cohen's general method, first studied in detail in my dissertation of 1968 and now known as Mathias forcing, with which I proved the consistency of a central principle of infinitary Ramsey theory.

B,C) The next two lists study further the two sides of the dual foundation mentioned above. The papers in List B examine systems, such as those proposed by the physicist Zermelo, the algebraist Mac Lane, and others whose interests are chiefly geometrical. [B2] and [B3] pinpoint weaknesses; [B1] and [B4] suggest ways of making these systems more ``recursion-friendly" without increasing their consistency strength; and [B1] includes a comparison of these systems with the type theory of Russell and Whitehead as simplified by Ramsey. For the study of mathematical induction and, more abstractly, recursion on well-founded relations, the power set axiom recedes in importance; and the focus of the papers in List C is on systems with no power set axiom at all. The paper [C1], by a study of extremely weak systems, identifies and corrects the flaws in Devlin's account of the theory of rudimentary functions, which theory forms the basis of Jensen's sweeping extension of Gödel's theory of constructibility. That task done, [C2] presents my theory of rudimentary recursion, incorporating improvements due to Nathan Bowler, which, as a first application, is shown in [C3] to form a sufficient basis for Cohen's theory of set forcing.

D) The papers in List D are concerned with strong axioms of infinity. Even if the physical universe is finite, we need infinity to simplify our picture of it; and sometimes, as discussed in [D6], we must go even further and appeal to higher orders of infinity to solve a problem. The collaborative papers [D7] and [D8] investigate the effect of large cardinals, as in set theory, on the behaviour of various functors, as in category theory.

E) Mathematics draws research projects from Natural Science: so it is natural to ask whether ordinals occur in Nature, and I was thus led, in my papers in List E, to investigate an iteration problem arising in symbolic dynamics. [E5] constructs a recursive point in Baire space where the said iteration continues for uncountably many stages.

List F contains a miscellany of papers, largely written for the pure pleasure of solving attractive problems.

List G contains discussions of various philosophies of mathematics and a campaign article.

List H contains lecture notes, survey articles and shorter expository pieces.

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