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On sets, games and processes

Cox, Michael (2014)
Ph.D. thesis, University of Birmingham.

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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:Ph.D. thesis.
Supervisor(s):Kaye, Richard
School/Faculty:Colleges (2008 onwards) > College of Engineering & Physical Sciences
Department:School of Mathematics
Subjects:QA Mathematics
Institution:University of Birmingham
ID Code:5077
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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