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Paul B. Larson a BRIEF HISTORY of DETERMINACY §1. Introduction
Paul B. Larson A BRIEF HISTORY OF DETERMINACY x1. Introduction. Determinacy axioms are statements to the effect that certain games are determined, in that each player in the game has an optimal strategy. The commonly accepted axioms for mathematics, the Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC; see [??, ??]), im- ply the determinacy of many games that people actually play. This applies in particular to many games of perfect information, games in which the players alternate moves which are known to both players, and the out- come of the game depends only on this list of moves, and not on chance or other external factors. Games of perfect information which must end in finitely many moves are determined. This follows from the work of Ernst Zermelo [??], D´enesK}onig[??] and L´aszl´oK´almar[??], and also from the independent work of John von Neumann and Oskar Morgenstern (in their 1944 book, reprinted as [??]). As pointed out by Stanis law Ulam [??], determinacy for games of perfect information of a fixed finite length is essentially a theorem of logic. If we let x1,y1,x2,y2,::: ,xn,yn be variables standing for the moves made by players player I (who plays x1,::: ,xn) and player II (who plays y1,::: ,yn), and A (consisting of sequences of length 2n) is the set of runs of the game for which player I wins, the statement (1) 9x18y1 ::: 9xn8ynhx1; y1; : : : ; xn; yni 2 A essentially asserts that the first player has a winning strategy in the game, and its negation, (2) 8x19y1 ::: 8xn9ynhx1; y1; : : : ; xn; yni 62 A essentially asserts that the second player has a winning strategy.1 We let ! denote the set of natural numbers 0; 1; 2;::: ; for brevity we will often refer to the members of this set as \integers". -
Arxiv:1801.09149V1 [Math.CA] 27 Jan 2018 .Byn H Osrcil Hierarchy Constructible the Beyond 5
Set Theory and the Analyst by N. H. Bingham and A. J. Ostaszewski Then to the rolling heaven itself I cried, Asking what lamp had destiny to guide Her little children stumbling in the dark. And ‘A blind understanding’ heaven replied. – The Rubaiyat of Omar Khayyam Abstract. This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure – category-measure duality and non-duality, as it were. The bulk of the text is devoted to a summary, intended for the working analyst, of the extensive background in set theory and logic needed to discuss such matters: to quote from the Preface of Kelley [Kel]: ”what every young analyst should know”. Table of Contents 1. Introduction 2. Early history 3. G¨odel Tarski and their legacy 4. Ramsey, Erd˝os and their legacy: infinite combinatorics 4a. Ramsey and Erd˝os 4b. Partition calculus and large cardinals 4c. Partitions from large cardinals 4d. Large cardinals continued arXiv:1801.09149v1 [math.CA] 27 Jan 2018 5. Beyond the constructible hierarchy L – I 5a. Expansions via ultrapowers 5b. Ehrenfeucht-Mostowski models: expansion via indiscernibles 6. Beyond the constructible hierarchy L – II 6a. Forcing and generic extensions 6b. Forcing Axioms 7. Suslin, Luzin, Sierpiński and their legacy: infinite games and large cardinals 7a. Analytic sets. 7b. Banach-Mazur games and the Luzin hierarchy 1 8. Shadows 9. The syntax of Analysis: Category/measure regularity versus practicality 10. Category-Measure duality Coda 1. Introduction An analyst, as Hardy said, is a mathematician habitually seen in the company of the real or complex number systems. -
Invisible Genericity and 0
Invisible Genericity and 0# M.C. Stanley February 1995 Abstract. 0# can be invisibly class generic. 1. Introduction . 1 1.1. Class genericity . 2 1.2. H(A) ............................. 4 2. Almost anything might be invisibly generic . 5 3. Two remarks . 12 4. The regularity of o(V) .......................16 5. Open problems . 21 5.1. Is the real 0# generic? . 21 5.2. Non-generic reals . 23 6. References . 23 1. Introduction Roughly, the main theorem of this paper is that some instances of any type of non- constructible object are class generic over L, in one sense. Since small large cardinal properties are inherited in L, let us begin by considering the sense in which 0# can be generic. It is well known that 0# is not set generic over L, or, indeed, over any inner model M such that 0# ∈/ M. This is because, if 0# ∈/ M, then unboundedly many M-cardinals are collapsed in L[0#], a trick no set forcing can perform. It is also known that there can be no L-definable class forcing P such that 0# ∈ L[G] and hL[G]; Gi ² ZFC, for some L-definably generic G ⊆ P. This is because it is known that in such a case the forcing relation on a cone of conditions rooted in G must be L-definable because hL[G]; L, P,Gi ² ZFC; but then ³ ´ ∃p ∃ι p °P “ˇι is an indiscernible” and Lι ² ϕ defines L-truth over L. Further, it is known that there is no L-amenable class forcing property P such that 0# ∈ L[G] and hL[G]; P,Gi ² ZFC, for some hL; Pi-definably generic G ⊆ P. -
Some Challenges for the Philosophy of Set Theory
SOME CHALLENGES FOR THE PHILOSOPHY OF SET THEORY JAMES CUMMINGS 1. Introduction In this paper my main goal is to elaborate on some ideas from a talk which I gave at the workshop on \Set Theory and the Philosophy of Mathematics" held at the University of Pennsylvania in October 2010. I will state and give supporting evidence for several theses about the philosophy of set theory and the Continuum Hypothesis (CH). Thesis I: (Really a general thesis about the philosophy of mathematics) A successful philosophy of set theory should be faithful to the mathe- matical practices of set theorists. In particular such a philosophy will require a close reading of the mathematics produced by set theorists, an understanding of the history of set theory, and an examination of the community of set theorists and its interactions with other mathe- matical communities. Note: I am not making any claim about the nature of mathematical truth. In particular I am not claiming (and do not in fact believe) that mathematical truth can be reduced either to the practices of individual mathematicians, or to the sociology of the mathematical community. I am also not claiming that the philosophy of mathematics should be purely descriptive, just that it can only get any traction on the main issues by staying closely engaged with actual mathematics. Thesis II: Set theory is a branch of mathematics. It just happens to be a branch in which issues of independence and consistency lie close to the surface, and in which the tools of logic have proven very powerful, but it is not (or not only) about logic. -
Set Theory and the Analyst
European Journal of Mathematics (2019) 5:2–48 https://doi.org/10.1007/s40879-018-0278-1 REVIEW ARTICLE Set theory and the analyst Nicholas H. Bingham1 · Adam J. Ostaszewski2 For Zbigniew Grande, on his 70th birthday, and in memoriam: Stanisław Saks (1897–1942), Cross of Valour, Paragon of Polish Analysis, Cousin-in-law Then to the rolling heaven itself I cried, Asking what lamp had destiny to guide Her little children stumbling in the dark. And ‘A blind understanding’ heaven replied. – The Rubaiyat of Omar Khayyam Received: 14 January 2018 / Revised: 15 June 2018 / Accepted: 12 July 2018 / Published online: 12 October 2018 © The Author(s) 2018 Abstract This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure—category-measure duality and non-duality, as it were. The bulk of the text is devoted to a summary, intended for the working analyst, of the extensive background in set theory and logic needed to discuss such matters: to quote from the preface of Kelley (General Topology, Van Nostrand, Toronto, 1995): “what every young analyst should know”. Keywords Axiom of choice · Dependent choice · Model theory · Constructible hierarchy · Ultraproducts · Indiscernibles · Forcing · Martin’s axiom · Analytical hierarchy · Determinacy and large cardinals · Structure of the real line · Category and measure Mathematics Subject Classification 03E15 · 03E25 · 03E30 · 03E35 · 03E55 · 01A60 · 01A61 B Adam J. Ostaszewski [email protected] Nicholas H. Bingham [email protected] 1 Mathematics Department, Imperial College, London SW7 2AZ, UK 2 Mathematics Department, London School of Economics, Houghton Street, London WC2A 2AE, UK 123 Set theory and the analyst 3 Contents 1 Introduction ............................................ -
Some Challenges for the Philosophy of Set Theory
Introduction A theorem of Shelah A theorem of Farah A theorem of Magidor and Shelah Conclusion Some challenges for the philosophy of set theory James Cummings CMU 21 March 2012 www.math.cmu.edu/users/jcumming/phil/efi_slides.pdf 1 / 51 Introduction A theorem of Shelah A theorem of Farah A theorem of Magidor and Shelah Conclusion Thesis I: (Really a general thesis about the philosophy of mathematics) A successful philosophy of set theory should be faithful to the mathematical practices of set theorists. In particular such a philosophy will require a close reading of the mathematics produced by set theorists, an understanding of the history of set theory, and an examination of the community of set theorists and its interactions with other mathematical communities. This talk represents an experiment in doing this kind of philosophy. 2 / 51 Introduction A theorem of Shelah A theorem of Farah A theorem of Magidor and Shelah Conclusion Thesis II: Set theory is a branch of mathematics. It happens to be a branch in which issues of independence and consistency lie close to the surface, and in which the tools of logic have proven very powerful, but it is not (or not only) about logic. As in any successful field of mathematics, progress in set theory comes from a dialectic between “internal” developments elaborating the central ideas, and “external” developments in which problems and ideas flow back and forth between set theory and other areas (for example topology, algebra, analysis, the theory of games). 3 / 51 Introduction A theorem of Shelah A theorem of Farah A theorem of Magidor and Shelah Conclusion Thesis III: Modern set theory uses a wide range of methods and discusses a wide range of objects. -
A BRIEF HISTORY of DETERMINACY §1. Introduction
0001 0002 0003 A BRIEF HISTORY OF DETERMINACY 0004 0005 0006 PAUL B. LARSON 0007 0008 0009 0010 x1. Introduction. Determinacy axioms are statements to the effect that 0011 certain games are determined, in that each player in the game has an optimal 0012 strategy. The commonly accepted axioms for mathematics, the Zermelo{ 0013 Fraenkel axioms with the Axiom of Choice (ZFC; see [Jec03, Kun83]), imply 0014 the determinacy of many games that people actually play. This applies in 0015 particular to many games of perfect information, games in which the 0016 players alternate moves which are known to both players, and the outcome 0017 of the game depends only on this list of moves, and not on chance or other 0018 external factors. Games of perfect information which must end in finitely 0019 many moves are determined. This follows from the work of Ernst Zermelo 0020 [Zer13], D´enesK}onig[K}on27]and L´aszl´oK´almar[Kal1928{29], and also 0021 from the independent work of John von Neumann and Oskar Morgenstern 0022 (in their 1944 book, reprinted as [vNM04]). 0023 As pointed out by Stanis law Ulam [Ula60], determinacy for games of 0024 perfect information of a fixed finite length is essentially a theorem of logic. 0025 If we let x1,y1,x2,y2,::: ,xn,yn be variables standing for the moves made by 0026 players player I (who plays x1,::: ,xn) and player II (who plays y1,::: ,yn), 0027 and A (consisting of sequences of length 2n) is the set of runs of the game 0028 for which player I wins, the statement 0029 0030 (1) 9x18y1 ::: 9xn8ynhx1; y1; : : : ; xn; yni 2 A 0031 essentially asserts that the first player has a winning strategy in the game, 0032 and its negation, 0033 0034 (2) 8x19y1 ::: 8xn9ynhx1; y1; : : : ; xn; yni 62 A 0035 essentially asserts that the second player has a winning strategy.1 0036 0037 The author is supported in part by NSF grant DMS-0801009. -
Remember, a Theory Is a Collection of Axioms and Logic)
BEYOND INFINITY 6 JADE NINE 1. Models A model is a collection of objects which satisfies a theory (remember, a theory is a collection of axioms and logic). The point of a model is to give a theory enough playing blocks and blueprints that it can satisfy all needs it possibly has. In set theory, a model is just a collection of sets where every set that an axiom says should exist, does exist. Here is a simple theory, which I will call the Theory of Steps: Axiom 1 (Existence). 9x Axiom 2 (Step up). 8x(9x+(x+ 6= x)) Axiom 3 (Step down). 8x(9x−(x− 6= x)) Axiom 4 (Navigation). 8x((x+)− = (x−)+ = x)) Can you think of a model for this theory? A set we have introduced in this course actually fulfills the requirements of a model for the Theory of Steps. The integers, Z, with the operations +; − defined as +1,−1, satisfies all the axioms: there exists an integer (the set is not empty), for each integer there is a x+ = x + 1 and a x− = x − 1, and the actions + and - are inverses of each other (they cancel each other out). Draw out a diagram for this model, and use the axioms to move around between the various points in the model. Notice that, no matter which axiom you pick, the destination stipulated by the axiom is available in the integers - for example, if you're at 5 and you want to move to 5+, you can move to 5 + 1 = 6. Can you think of any other models? In fact, two disjoint (separate) copies of the integers also satisfies the theory. -
Introduction 3 by Akihiro Kanamori 1 Beginnings
Contents 0. Introduction 3 by Akihiro Kanamori 1 Beginnings . 4 1.1 Cantor . 4 1.2 Zermelo . 6 1.3 First Developments . 7 1.4 Replacement and Foundation . 11 2 New Groundwork . 15 2.1 G¨odel . 15 2.2 Infinite Combinatorics . 18 2.3 Definability . 21 2.4 Model-Theoretic Techniques . 23 3 The Advent of Forcing . 28 3.1 Cohen . 28 3.2 Method of Forcing . 30 3.3 0#, L[U], and L[U] . 35 3.4 Constructibility . 38 4 Strong Hypotheses . 42 4.1 Large Large Cardinals . 42 4.2 Determinacy . 45 4.3 Silver's Theorem and Covering . 49 4.4 Forcing Consistency Results . 53 5 New Expansion . 57 5.1 Into the 1980s . 57 5.2 Consistency of Determinacy . 62 5.3 Later Developments . 65 6 Summaries of the Handbook Chapters . 69 1 2 CONTENTS 0. Introduction Akihiro Kanamori Set theory has entered its prime as an advanced and autonomous research field of mathematics with broad foundational significance, and this Handbook with its expanse and variety amply attests to the fecundity and sophistication of the subject. Indeed, in set theory's further reaches one sees tremendous progress both in its continuing development of its historical heritage, the investigation of the transfinite numbers and of definable sets of reals, as well as its analysis of strong propositions and consistency strength in terms of large cardinal hypotheses and inner models. This introduction provides a historical and organizational frame for both modern set theory and this Handbook, the chapter summaries at the end be- ing a final elaboration. -
Extending the Language of Set Theory
Dmytro Taranovsky April 20, 2005 Extending the Language of Set Theory Note (March 10, 2012): A follow-up paper on higher order theory is now available: "Reflective Cardinals", arXiv:1203.2270. Abstract: We discuss the problems of incompleteness and inexpressibility. We introduce almost self-referential formulas, use them to extend set theory, and relate their expressive power to that of infinitary logic. We discuss the nature of proper classes. Finally, we introduce and axiomatize a powerful extension to set theory. Contents: Introduction and Outline, Incompleteness and Inexpressibility, Almost Self-Referential Formulas, Expressive Power of the Extensions, A Hierarchy of Large Cardinals, Proper Classes, Higher Order Set Theory. Update (December 19, 2016): Added references and revised two sections ("Almost Self-Referential Formulas" and "Expressive Power of the Extensions"); content updates in other sections are marked with 'update'. Introduction and Outline Despite vast advances in set theory and mathematics in general, the language of set theory, which is also the language of mathematics, has remained the same since the beginning of modern set theory and first order logic. That formal language has served us well, but it is of necessity limited, and does not adequately deal with properties for which there is no set of all objects satisfying the property. The purpose of this paper is to address the deficiencies in the expressive power of ordinary set theory. We assume general familiarity with mathematical logic and set theory. A good exposition of set theory is in [1] or [2], the later having more emphasis on large cardinals, including history and philosophy. The second section discusses incompleteness and inexpressibility in general. -
A Transcendental View on the Continuum: Woodin's Conditional
A transcendental view on the continuum: Woodin’s conditional platonism Jean PETITOT∗ December 2007 To appear in Intellectica. Abstract One of the main difficulty concerning the nature of the continuum is to do justice, inside the set theoretical Cantorian framework, to the classical conception (from Aristotle to Thom, via Kant, Peirce, Brentano, Husserl and Weyl) according to which the continuum is a non-compositional, cohesive, primitive, and intuitive datum. This pa- per investigates such possibilities, from G¨odel to Woodin, of modelling inside a ZFC -universe the transcendence of the intuitive continuum w.r.t. its symbolic determination. Keywords: constructive universe, continuum, G¨odel, forcing, Kant, large cardinals, Ω-logic, projective hierarchy, V = L, Woodin, 0#. 1 Introduction One of the main general philosophical problem raised by the nature of the continuum is the conflict between two traditions: the older one, which can be called “neo-Aristotelian”, even if this sounds rather vague, and the now the classical one, namely the Cantorian tradition. arXiv:1502.05898v1 [math.LO] 20 Feb 2015 According to the neo-Aristotelian tradition, the continuum is experi- enced and thought of as a non-compositional, cohesive, primitive, and ∗EHESS & CREA (Ecole Polytechnique), France. [email protected]. 1 intuitive datum. It can be segmented into parts but these parts are themselves continua and points are only their boundaries. This point of view was very well defended by Kant. As soon as in his 1770 Dissertatio, he emphasized the fact that “a magnitude is continuous when it is not composed out of simple elements” (AK, II, p. 399 1), and explained that for the continuous “pure intuitions” of space and time “any part of time is still a time, and the simple elements which are in time, namely the moments, are not parts but limits between which a time takes place” (AK, II, p. -
Measurable Cardinal Wikipedia Contents
Measurable cardinal Wikipedia Contents 1 Atom (measure theory) 1 1.1 Definition ............................................... 1 1.2 Examples ............................................... 1 1.3 Non-atomic measures ......................................... 1 1.4 See also ................................................ 2 1.5 Notes ................................................. 2 1.6 References ............................................... 2 2 Axiom of determinacy 3 2.1 Types of game that are determined .................................. 3 2.2 Incompatibility of the axiom of determinacy with the axiom of choice ................ 3 2.3 Infinite logic and the axiom of determinacy .............................. 4 2.4 Large cardinals and the axiom of determinacy ............................ 4 2.5 See also ................................................ 4 2.6 References ............................................... 5 2.7 Further reading ............................................ 5 3 Cardinal number 6 3.1 History ................................................. 6 3.2 Motivation .............................................. 8 3.3 Formal definition ........................................... 9 3.4 Cardinal arithmetic .......................................... 10 3.4.1 Successor cardinal ...................................... 10 3.4.2 Cardinal addition ....................................... 10 3.4.3 Cardinal multiplication .................................... 11 3.4.4 Cardinal exponentiation ................................... 11 3.5