ON SETS, GAMES AND PROCESSES by MICHAEL COX A thesis submitted to University of Birmingham for the degree of DOCTOR OF PHILOSOPHY Mathematics College of Engineering and Physical Sciences University of Birmingham July 2013 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties.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. 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 de- veloped. 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 moti- vations 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. For Bill. ACKNOWLEDGEMENTS Generous thanks are due to my family, for their support and persistence with my educa- tion, without which I would not have come nearly so far. Also to Sandi, for her enduring support and encouragement. Great thanks are of course due to my Ph.D. and Masters supervisor, Dr. Richard Kaye. He has provided fresh insight and is a constant source of ideas. All material presented here should be considered joint work with him. Also I am grateful to my fellow Ph.D. students and the staff at the university; I must mention Tin Lok Wong, who produced the bhamthesis class for LATEX, which made formatting this thesis bearable. CONTENTS 1 Introduction 1 1.1 Brief survey of relevant literature . 1 1.1.1 Combinatorial game theory . 1 1.1.2 Topological set theory . 11 1.2 This thesis . 14 1.2.1 Amphi-ZF . 14 1.2.2 Order theory of combinatorial games . 14 1.2.3 Extensions in category theory . 15 1.2.4 Topological set theory in nonstandard arithmetic . 17 2 A two-sided set theory for combinatorial games 18 2.1 Preliminaries . 19 2.2 Amphi-ZF . 21 2.3 Interpreting Amphi-ZF in ZF . 26 2.4 Interpreting ZF in Amphi-ZF . 29 2.5 Amphi-NBG . 33 2.6 Rieger{Bernays permutation models . 35 2.7 Open questions and suggestions for future research . 41 3 Two-ordered structures 43 3.1 Two-ordered structures . 44 3.1.1 Definitions and examples . 44 3.1.2 Morphisms of two-ordered structures . 47 3.1.3 Two-orders and boolean algebras . 49 3.2 Structures of two-orders . 52 3.2.1 Extending and reducing two-orders . 52 3.2.2 A representation theorem for posets . 56 3.3 Two-ordered groups . 60 3.3.1 Morphisms and quotients . 64 3.3.2 Homomorphism theorems . 66 3.3.3 Automorphisms of two-ordered structures . 72 3.4 Duality and determinacy . 74 4 Game categories 78 4.1 Preliminaries . 78 4.2 Game categories . 79 4.2.1 Functors of game categories . 80 4.3 Extending and reducing game categories . 86 4.4 Products of game categories . 90 4.4.1 Products in GC ............................. 90 4.4.2 Products in GCam ........................... 93 4.5 Enriched game categories and the value map . 95 4.6 Monoidal game categories . 99 4.6.1 Monoidal game categories and the value map . 101 4.7 Architectures . 102 4.7.1 Architectures and the value map . 105 4.8 Gamuts . 106 4.8.1 Examples of gamuts . 107 5 2-architectures 112 5.1 Building 2-architectures from amphisets . 113 5.1.1 The least order method as a functor . 120 5.1.2 The dual of T m .............................121 5.2 Extending existing architectures . 122 5.3 Cuts in Peano arithmetic as surreal numbers over a model . 125 5.3.1 Rifts as a measure of size . 134 5.4 Architectures as a threading model . 135 6 Topological set theory and nonstandard arithmetic 138 6.1 Preliminaries . 138 6.1.1 Properties of cuts . 139 6.2 The construction . 141 6.2.1 Sequences . 142 6.3 Positive set theory in V .............................144 6.4 Reverse results . 147 6.4.1 Crowdedness . 147 6.4.2 Completeness . 149 6.4.3 Witnessing principles . 150 6.5 The relative strength of CDF . 155 7 Conclusions 159 7.1 In game theory . 159 7.2 In nonstandard topological set theory . 160 List of References 161 CHAPTER 1 INTRODUCTION This thesis reports the more successful research from a period of study between 2009 and 2013; some results here beg for improvement and are unlikely to be optimal. Chapters 2-5 in particular build upon the author's MPhil(Qual) thesis, Contributions to the theory of combinatorial games [16]. As such some material should be recognisable to a reader of the MPhil(Qual), but also more complete. Chapter 6 is completely new, although it too has some roots in the predecessor. That chapter is also somewhat unrelated to the rest of the thesis, but certainly reflects an important part of the author's PhD studies. We begin with a brief survey of relevant literature. This will help us to put the current work into context, and also is an ideal place to introduce necessary concepts and notation. 1.1 Brief survey of relevant literature 1.1.1 Combinatorial game theory Game theory is a vast subject area, and as such it has been necessary throughout to restrict attention to a very small subset. This was achieved by considering primarily combinatorial games1, or two-player games of perfect information, in which players alternate in taking 1By combinatorial game we will always mean a two-player game of perfect information, where the players alternate turns. Often these will come with some notion of strategy, and moving from one position to another. Positions tend to be identified with games (so that each position in a game is considered a game in its own right, and each game may be considered a position in a larger game). 1 their turns. Although some of the research below will apply outside this restricted area, our focus will only be on such games. Impartial games, Nim, and the Sprague-Grundy theorem The first significant result of combinatorial game theory was the Sprague-Grundy theorem, discovered independently by R.P. Sprague [81] and P.M. Grundy [42]. This concerned impartial combinatorial games: games between two players, where each move available to one is available to both, and in which the players alternate to take turns and the last to move wins. One particularly distinguished such game is Nim, in which players take turns to remove (for instance) counters from a number of stacks. The initial state or position of the game is some configuration involving any number (0; 1; 2;:::) of heaps of counters. A player must, on their turn, select one heap and remove any positive number of counters from it. Under the normal play condition, the last player to remove any counters is the winner (so whichever player cannot make a move when required to loses, and the game terminates). The outcomes of Nim positions are equivalent to nimbers, defined as follows. The empty game (having no moves for either player, and hence a definite win for the second player, II), say fg, is the nimber ∗0. We then set ∗1 = f∗0g, the game in which each player may move only to the position ∗0 (hence this position is a definite win for the first player). We define ∗2 = f∗0; ∗1g; . ∗(n + 1) = f∗0; ∗2;:::; ∗ng; . Assuming that the two players play according to the best possible strategies (in the finite, wellfounded impartial games considered by Sprague and Grundy there is always 2 an optimal strategy), we can talk of the outcome class of a position G: say G is an N - position if the next player to move (i.e. I) wins, and a P-position if the previous player to move (i.e. II) wins. For wellfounded such games there is always a winner, and hence every impartial, wellfounded combinatorial game is either an N -position or a P-position. In particular, ∗0 is a P-position, while every other nimber (allowing the first player to move to ∗0 and hence to win) is an N -position. If G; H are games then we may form the disjunctive sum G + H by allowing each player, on their turn, to pick exactly one of G; H and proceed by moving in that game as usual.