Some axioms of weak determinacy [Elektronische Ressource] / vorgelegt von Bogomil Kovachev

Bogomil Kovachev

Description

Some Axioms of Weak DeterminacyDissertationan der Fakult at fur Mathematik, Informatik und Statistikder Ludwig Maximilians-Universit at Munc henzur Erlangung des Grades Doctor rerum naturalium (Dr. rer. nat.)vorgelegt von Bogomil Kovachev (23. Juni 2009)Erster Berichterstatter: Prof. Dr. H.-D. DonderZweiter Berichterstatter: Prof. Dr. P.KoepkeTag des Rigorosums: 16.10.2009AbstractWe consider two-player games of perfect information of length some car-dinal . It is well-known that for ! the full axiom of determinacy for1these games fails, thus we investigate three weaker forms of it. We obtain+the measurability of under DC -the axiom of dependent choices general-ized to . We generalize the notions of perfect and meager sets and providecharacterizations with some special kinds of games. We show that under anadditional assumption one of our three axioms follows from the other two.ZusammenfassungWir betrachten unendliche Spiele der L ange , wobei eine Kardinalzahlist. Es ist bekannt, dass im Fall ! fur solche Spiele das ublic he De-1terminierheitsaxiom inkonsistent ist. Aus diesem Grund betrachten wir dreischw achere Versionen hiervon. Mit hilfe von DC zeigen wir die Messbarkeit+von . Wir verallgemeinern die bekannten Begri e der perfekten undmageren Mengen und geben Charakterisierungen durch spezielle Spielvari-anten.

Sujets

• Mathematics

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2009
• Nombre de lectures: 9
• Langue: English

Extrait

Some Axioms of Weak Determinacy

Dissertation anderFakult¨atf¨urMathematik,InformatikundStatistik derLudwigMaximilians-Universita¨tMu¨nchen zur Erlangung des Grades Doctor rerum naturalium (Dr. rer. nat.)

vorgelegt von Bogomil Kovachev (23. Juni 2009)

Erster Berichterstatter: Prof. Dr. H.-D. Donder Zweiter Berichterstatter: Prof. Dr. P.Koepke Tag des Rigorosums: 16.10.2009

Abstract

We consider two-player games of perfect information of length some car- dinalκ. It is well-known that forκ≥ω 1 the full axiom of determinacy for these games fails, thus we investigate three weaker forms of it. We obtain the measurability ofκ + underDC κ -the axiom of dependent choices general- ized toκnotions of perfect and meager sets and provide generalize the . We characterizations with some special kinds of games. We show that under an additional assumption one of our three axioms follows from the other two.

Zusammenfassung

WirbetrachtenunendlicheSpielederLa¨ngeκ, wobeiκeine Kardinalzahl ist. Es ist bekannt, dass im Fallκ≥ω 1 pielefd¨ausrsolcheSeD-u¨lbcieh terminierheitsaxiom inkonsistent ist. Aus diesem Grund betrachten wir drei schwa¨chereVersionenhiervon.MithilfevonDC κ zeigen wir die Messbarkeit vonκ +  verallgemeinern die bekannten Begriffe der perfekten und. Wir mageren Mengen und geben Charakterisierungen durch spezielle Spielvari- anten.Untereinerzus¨atzlichenVoraussetzungzeigenwir,dasseinsunserer drei Axiomen aus den anderen beiden folgt.

Contents 1 Introduction 3 2 The spaces κ κand κ 26 3 Trees and perfect sets 9 4 Turing reducibility forC κ andB κ 14 5 Games of lengthκ21 5.1 Games and Strategies . . . . . . . . . . . . . . . . . . . . . . . 21 5.1.1 The gameG κ κ (A . . . . . . . . . . . . . . . . . . . . 21) . 5.1.2 The gameG κ 2 (A) . . . . . . . . . . . . . . . . . . . . . 25 5.1.3 The gamesG ∗∗ κ κ (A) andG ∗ κ κ (A 27) . . . . . . . . . . . . 5.2 Some observations . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2.1 Finite games . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2.2 Games of lengthω. . . . . . . . .. . . . . . . . . . .  29 5.2.3 Games of lengthκ. . . . . . . . . . . . . . . . . . . . 29 6 The Axioms Of Weak Determinacy 32 6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2∗and∗∗determinacy .. . . . . . . . . . . . . . . . . . . . . .  34 6.3 Turing determinacy . . . . . . . . . . . . . . . . . . . . . . . . 36 7 Consequences 41 7.1 Measurability ofκ +  41. . . . . . . . . . . . . . . . . . . . . . . . 7.2∗ 43. . . . . . . . . . . -determinacy and the perfect set property 7.3∗∗-determinacy andκ 45 . . . . . . . . . . . . . . .-meager sets . 8 A proof of∗-determinacy from Turing determinacy and∗∗- determinacy 48 8.1 The modelW κ . . . . . . . . . . . . . . . . . . . . . . . . . . 48 8.2 The main proof . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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1 Introduction

The Axiom of Determinacy is the following statement: (AD)Every two-person game of lengthωwhere the players play ordinals smaller thanωis determined. A natural question to ask is whether the restriction on the length could be relaxed, i.e. whether an axiom of the following (stronger) kind is worth considering: (AD κ )Every two-person game of lengthκwhere the players play ordinals smaller thanκis determined. It turns out that, forκ≥ω 1 ,AD κ is inconsistent with the axioms of ZF. The topic of this dissertation are three ways of weakeningAD κ which might be consistent with ZF,DC κ andκ= 2 <κ . These three axioms are well-known and studied in the caseκ=ωunder the names Turing Determinacy (TD), ∗-Determinacy and∗∗-Determinacy. Informally our generalizations will be as follows. The axiomT D κ says roughly that if we define Turing degrees onκ-sequences in a natural way, then the union of every set of Turing degrees is determined as a subset of the set of allκ axiom of-sequences. The∗-Determinacy just states that if player Iis allowed to play bounded subsets ofκ, then the game is determined and ∗∗the same under the condition that both players are-determinacy states allowed to play non-empty bounded subsets. We will study some consequences of these axioms in the presence of the axiomDC κ which is a generalization of the usual axiom of dependent choices. With the help of Turing determinacy we present a generalization of the result of Martin that every set of Turing degrees contains a cone or is disjoint from a cone and thus show the measurability of the ”next cardinal , while with the ” help of∗determinacy we provide a generalization of the perfect set property for the reals, whose role is played byκ-sequences in our context.

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