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The Modal Logic of Potential Infinity, with an Application to Free Choice

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The Modal Logic of Potential Infinity, with an Application to Free Choice The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ethan Brauer, B.A. ∼6 6 Graduate Program in Philosophy The Ohio State University 2020 Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument|which is due to Linnebo and which I partially endorse|that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity. I argue that there is an important range of potential infinities, which I call infinities with branching possibilities, for which S4 is the right modal logic. I further argue that the usual operator is not sufficiently expressive for reasoning about branching potential infinities. A new operator is needed, which I call the inevitability operator. (In tense logic, the same operator has been called the strong future tense). I show that first-order S4 with the inevitability operator is ii unaxiomatizable. This makes it improbable that there can be a faithful translation of non-modal mathematical discourse into modal language. I suggest that this means standard mathematical theories of infinite sequences are thus committed to actual infinities. In Part II, I apply the modal account of potential infinity to develop a theory of free choice sequences. Free choice sequences are a concept from intuitionistic mathe- matics. They can be thought of as potentially infinite sequences of natural numbers whose values are freely chosen one at a time by an (idealized) mathematician. They figure prominently in results of intuitionistic mathematics that contradict classical mathematics. I propose an axiomatic theory of free choice sequences in a modal ex- tension of classical second-order arithmetic, called MC, with the aim of providing modal analogues of key ideas and results from the intuitionistic analysis. In this the- ory I define the temporal-potential continuum, which serves as an ersatz intuitionistic continuum in my modal theory. I show that the temporal-potential continuum exhibits many characteristically intuitionistic properties. The main results are these: it is not the case that every real number is determinately rational or irrational; the natural order on real numbers is not linear; a bounded monotone sequence of rationals need not be Cauchy; if two disjoint sets A and B decompose the temporal-potential continuum, then both A and B are topologically open; finally, I introduce a notion of sharp discontinuity and show there is no function on the temporal-potential continuum which is sharply discontinuous. iii Acknowledgments Many thanks are due to my dissertation committee, Stewart Shapiro, Neil Tennant, Chris Miller, and Chris Pincock. They have been most helpful in discussing and commenting on multiple drafts of papers, including several which did not end up in this dissertation. The resulting work would have been much the worse if not for their help. I especially appreciate that Neil and Stewart have spent so much time discussing the rough and vague ideas that I have at one time or another thought about pursuing. A good portion of this dissertation was written in the summer of 2019 during a stay at the University of Oslo. Thanks to Øystein Linnebo, who graciously spent many hours talking about this material as I worked on it. I presented the material in Chapter 3 at the 2020 Cambridge Graduate Conference on the Philosophy of Math- ematics and Logic. Thanks to Wes Wrigley for comments on that occasion and to Robin Solberg for helpful conversation. Material from chapters 4 to 6 was presented to the Ohio State Logic Seminar and a University of Connecticut Logic Colloquium in the fall of 2019; thanks to audiences on both occasions for helpful questions and comments, especially Tim Carlson, Neil DeBoer, Ivo Herzog, Marcus Rossberg, and Lionel Shapiro. Many teachers have been influential in my growth as a philosopher, and without a doubt they have had an impact on this work, even if it is only felt indirectly. In this connection I want to thank especially Ben Caplan, Jeff Dunn, Marcia McKelligan, and Declan Smithies. A debt is also owed to my fellow graduate students at Ohio State iv and I would be remiss not to mention the members of the Logic or Language Society from whom I have learned much: Andr´eCurtis-Trudel, Steven Dalglish, Teresa Kouri Kissel, Giorgio Sbardolini, and Damon Stanley. Equally important are those whose company over the last several years has been as likely to involve beer as philosophy: Rachel Harris, Preston Lennon, Erin Mercurio, Daniel Olson, and Evan Woods. The largest, if least tangible, debt is due to my family. Thanks to my parents Tom and Janice, my brother Matt and sister-in-law Hallie, and finally to my grandmother Mary to whom this work is dedicated. v Vita 2014 . B.A. Physics and Philosophy, DePauw University 2014-2020 . Graduate Teaching Associate, Dept. of Philosophy, Ohio State University Publications \Second-order Logic and the Power Set," 2018. Journal of Philosophical Logic Vol. 47 No. 1, pp. 123-142. \Relevance for the Classical Logician," 2018. Review of Symbolic Logic, forthcoming. Fields of Study Major Fields: Philosophy vi Contents Abstract ...................................................................... ii Acknowledgments ............................................................. iv Vita ........................................................................... vi List of Figures ................................................................. ix 1 Introduction ............................................................... 1 1.1 Modal Logic................................7 1.2 Free Choice Sequences.......................... 13 I The Modal Logic of Potentialism .................... 18 2 Convergent Possibilities .................................................... 19 2.1 The Role of the Mirroring Theorems.................. 19 2.2 The Need for Free Logic......................... 26 2.3 Classical Free Logic............................ 28 2.4 Intuitionistic Free Logic......................... 34 2.5 Conclusion................................. 38 3 Branching Possibilities ..................................................... 40 3.1 Hamkins on Potentialism......................... 42 3.2 Branching vs. Convergent Possibilities................. 44 3.3 An Incompleteness Theorem....................... 50 3.4 Axioms for Inevitability......................... 58 3.5 The Philosophical Upshot........................ 59 3.6 A Remark on Models and Potentialism................. 62 3.7 Conclusion................................. 65 II A Modal Theory of Free Choice Sequences ......... 66 vii 4 Free Choice Sequences ..................................................... 67 4.1 The goal.................................. 70 4.2 Free choice sequences........................... 72 4.3 The modal approach........................... 75 4.3.1 Notational preliminaries..................... 76 4.3.2 Conceptions of choice sequences................. 77 5 Lawless Sequences ......................................................... 83 5.1 The Intuitionistic Theory of Lawless Sequences............ 83 5.2 The Modal Theory of Lawless Sequences................ 87 5.2.1 A Kripke Model for MCLS .................... 92 5.2.2 Translations into MCLS ..................... 95 5.2.3 Basic Properties of MCLS .................... 101 5.2.4 Continuity Principles....................... 105 5.2.5 The Bar Theorem......................... 113 5.3 Conclusion................................. 116 6 Non-Lawless Sequences and Real Numbers .................................118 6.1 A Kripke Model for MC ......................... 123 6.2 Kripke's Schema............................. 126 6.3 Bar Induction and the Fan Theorem.................. 128 6.4 Real numbers............................... 131 6.5 Continuity................................. 143 6.6 Decomposing the Continuum....................... 148 6.7 Conclusion................................. 164 References .....................................................................166 Appendices ....................................................................173 A Frame Conditions For Intuitionistic S4.2 ...................................173 B The Need for Axiom S8 ....................................................175 C More Stability Properties ..................................................178
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