On sets, games and processes

Cox, Michael (2014). On sets, games and processes. University of Birmingham. Ph.D.

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Abstract

We introduce a two-sided set theory, Amphi-ZF, based on the pure games of Conway et al.; we show Amphi-ZF and ZF are synonymous, with the same result for important subtheories.

An order-theoretic generalisation of Conway games is introduced, and the theory developed. We show the collection of such orders over a poset possesses rich structure, and an analogue of Stone's theorem is proved for posets, using these spaces.

These generalisations are then considered using categories. Compatible set-theoretic notions are introduced, and ideas of regularity axioms with purely game-theoretic motivations are explored; applications to nonstandard arithmetic and multithreaded software are proposed.

We consider topological set theory in a nonstandard model M of Peano arithmetic, and demonstrate that Malitz' original construction works in a finite set theory interpreted by M, with the usual cardinal replaced by a special initial segment. This gives a suitably compact topological model of GPK. Reverse results are also considered: crowdedness of the topological model holds iff the initial segment is strong. A reverse-mathematical principle is investigated, and used it to show that completeness of the topological model is much weaker. Comparisons are made with the standard situation as investigated by Forti et al.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):