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Calibrating Determinacy Strength in Levels of the Borel Hierarchy
CALIBRATING DETERMINACY STRENGTH IN LEVELS OF THE BOREL HIERARCHY SHERWOOD J. HACHTMAN Abstract. We analyze the set-theoretic strength of determinacy for levels of the Borel 0 hierarchy of the form Σ1+α+3, for α < !1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to require α+1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of Π1-reflection principles, Π1-RAPα, whose consistency strength corresponds 0 CK exactly to that of Σ1+α+3-Determinacy, for α < !1 . This yields a characterization of the levels of L by or at which winning strategies in these games must be constructed. When α = 0, we have the following concise result: the least θ so that all winning strategies 0 in Σ4 games belong to Lθ+1 is the least so that Lθ j= \P(!) exists + all wellfounded trees are ranked". x1. Introduction. Given a set A ⊆ !! of sequences of natural numbers, consider a game, G(A), where two players, I and II, take turns picking elements of a sequence hx0; x1; x2;::: i of naturals. Player I wins the game if the sequence obtained belongs to A; otherwise, II wins. For a collection Γ of subsets of !!, Γ determinacy, which we abbreviate Γ-DET, is the statement that for every A 2 Γ, one of the players has a winning strategy in G(A). It is a much-studied phenomenon that Γ -DET has mathematical strength: the bigger the pointclass Γ, the stronger the theory required to prove Γ -DET. -
Topology and Descriptive Set Theory
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector TOPOLOGY AND ITS APPLICATIONS ELSEVIER Topology and its Applications 58 (1994) 195-222 Topology and descriptive set theory Alexander S. Kechris ’ Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Received 28 March 1994 Abstract This paper consists essentially of the text of a series of four lectures given by the author in the Summer Conference on General Topology and Applications, Amsterdam, August 1994. Instead of attempting to give a general survey of the interrelationships between the two subjects mentioned in the title, which would be an enormous and hopeless task, we chose to illustrate them in a specific context, that of the study of Bore1 actions of Polish groups and Bore1 equivalence relations. This is a rapidly growing area of research of much current interest, which has interesting connections not only with topology and set theory (which are emphasized here), but also to ergodic theory, group representations, operator algebras and logic (particularly model theory and recursion theory). There are four parts, corresponding roughly to each one of the lectures. The first contains a brief review of some fundamental facts from descriptive set theory. In the second we discuss Polish groups, and in the third the basic theory of their Bore1 actions. The last part concentrates on Bore1 equivalence relations. The exposition is essentially self-contained, but proofs, when included at all, are often given in the barest outline. Keywords: Polish spaces; Bore1 sets; Analytic sets; Polish groups; Bore1 actions; Bore1 equivalence relations 1. -
Abstract Determinacy in the Low Levels of The
ABSTRACT DETERMINACY IN THE LOW LEVELS OF THE PROJECTIVE HIERARCHY by Michael R. Cotton We give an expository introduction to determinacy for sets of reals. If we are to assume the Axiom of Choice, then not all sets of reals can be determined. So we instead establish determinacy for sets according to their levels in the usual hierarchies of complexity. At a certain point ZFC is no longer strong enough, and we must begin to assume large cardinals. The primary results of this paper are Martin's theorems that Borel sets are determined, homogeneously Suslin sets are determined, and if a measurable cardinal κ exists then the complements of analytic sets are κ-homogeneously Suslin. We provide the necessary preliminaries, prove Borel determinacy, introduce measurable cardinals and ultrapowers, and then prove analytic determinacy from the existence of a measurable cardinal. Finally, we give a very brief overview of some further results which provide higher levels of determinacy relative to larger cardinals. DETERMINACY IN THE LOW LEVELS OF THE PROJECTIVE HIERARCHY A Thesis Submitted to the Faculty of Miami University in partial fulfillment of the requirements for the degree of Master of Arts Department of Mathematics by Michael R. Cotton Miami University Oxford, Ohio 2012 Advisor: Dr. Paul Larson Reader: Dr. Dennis K. Burke Reader: Dr. Tetsuya Ishiu Contents Introduction 1 0.1 Some notation and conventions . 2 1 Reals, trees, and determinacy 3 1.1 The reals as !! .............................. 3 1.2 Determinacy . 5 1.3 Borel sets . 8 1.4 Projective sets . 10 1.5 Tree representations . 12 2 Borel determinacy 14 2.1 Games with a tree of legal positions . -
A List of Arithmetical Structures Complete with Respect to the First
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 257 (2001) 115–151 www.elsevier.com/locate/tcs A list of arithmetical structures complete with respect to the ÿrst-order deÿnability Ivan Korec∗;X Slovak Academy of Sciences, Mathematical Institute, Stefanikovaà 49, 814 73 Bratislava, Slovak Republic Abstract A structure with base set N is complete with respect to the ÿrst-order deÿnability in the class of arithmetical structures if and only if the operations +; × are deÿnable in it. A list of such structures is presented. Although structures with Pascal’s triangles modulo n are preferred a little, an e,ort was made to collect as many simply formulated results as possible. c 2001 Elsevier Science B.V. All rights reserved. MSC: primary 03B10; 03C07; secondary 11B65; 11U07 Keywords: Elementary deÿnability; Pascal’s triangle modulo n; Arithmetical structures; Undecid- able theories 1. Introduction A list of (arithmetical) structures complete with respect of the ÿrst-order deÿnability power (shortly: def-complete structures) will be presented. (The term “def-strongest” was used in the previous versions.) Most of them have the base set N but also structures with some other universes are considered. (Formal deÿnitions are given below.) The class of arithmetical structures can be quasi-ordered by ÿrst-order deÿnability power. After the usual factorization we obtain a partially ordered set, and def-complete struc- tures will form its greatest element. Of course, there are stronger structures (with respect to the ÿrst-order deÿnability) outside of the class of arithmetical structures. -
The Infinite and Contradiction: a History of Mathematical Physics By
The infinite and contradiction: A history of mathematical physics by dialectical approach Ichiro Ueki January 18, 2021 Abstract The following hypothesis is proposed: \In mathematics, the contradiction involved in the de- velopment of human knowledge is included in the form of the infinite.” To prove this hypothesis, the author tries to find what sorts of the infinite in mathematics were used to represent the con- tradictions involved in some revolutions in mathematical physics, and concludes \the contradiction involved in mathematical description of motion was represented with the infinite within recursive (computable) set level by early Newtonian mechanics; and then the contradiction to describe discon- tinuous phenomena with continuous functions and contradictions about \ether" were represented with the infinite higher than the recursive set level, namely of arithmetical set level in second or- der arithmetic (ordinary mathematics), by mechanics of continuous bodies and field theory; and subsequently the contradiction appeared in macroscopic physics applied to microscopic phenomena were represented with the further higher infinite in third or higher order arithmetic (set-theoretic mathematics), by quantum mechanics". 1 Introduction Contradictions found in set theory from the end of the 19th century to the beginning of the 20th, gave a shock called \a crisis of mathematics" to the world of mathematicians. One of the contradictions was reported by B. Russel: \Let w be the class [set]1 of all classes which are not members of themselves. Then whatever class x may be, 'x is a w' is equivalent to 'x is not an x'. Hence, giving to x the value w, 'w is a w' is equivalent to 'w is not a w'."[52] Russel described the crisis in 1959: I was led to this contradiction by Cantor's proof that there is no greatest cardinal number. -
The Reverse Mathematics of Cousin's Lemma
VICTORIAUNIVERSITYOFWELLINGTON Te Herenga Waka School of Mathematics and Statistics Te Kura Matai Tatauranga PO Box 600 Tel: +64 4 463 5341 Wellington 6140 Fax: +64 4 463 5045 New Zealand Email: sms-offi[email protected] The reverse mathematics of Cousin’s lemma Jordan Mitchell Barrett Supervisors: Rod Downey, Noam Greenberg Friday 30th October 2020 Submitted in partial fulfilment of the requirements for the Bachelor of Science with Honours in Mathematics. Abstract Cousin’s lemma is a compactness principle that naturally arises when study- ing the gauge integral, a generalisation of the Lebesgue integral. We study the ax- iomatic strength of Cousin’s lemma for various classes of functions, using Fried- arXiv:2011.13060v1 [math.LO] 25 Nov 2020 man and Simpson’s reverse mathematics in second-order arithmetic. We prove that, over RCA0: (i) Cousin’s lemma for continuous functions is equivalent to the system WKL0; (ii) Cousin’s lemma for Baire 1 functions is at least as strong as ACA0; (iii) Cousin’s lemma for Baire 2 functions is at least as strong as ATR0. Contents 1 Introduction 1 2 Integration and Cousin’s lemma 5 2.1 Riemann integration . .5 2.2 Gauge integration . .6 2.3 Cousin’s lemma . .7 3 Logical prerequisites 9 3.1 Computability . .9 3.2 Second-order arithmetic . 12 3.3 The arithmetical and analytical hierarchies . 13 4 Subsystems of second-order arithmetic 17 4.1 Formal systems . 17 4.2 RCA0 .......................................... 18 4.3 WKL0 .......................................... 19 4.4 ACA0 .......................................... 21 4.5 ATR0 .......................................... 22 1 4.6 P1-CA0 ......................................... 23 5 Analysis in second-order arithmetic 25 5.1 Number systems . -
Descriptive Set Theory
Descriptive Set Theory David Marker Fall 2002 Contents I Classical Descriptive Set Theory 2 1 Polish Spaces 2 2 Borel Sets 14 3 E®ective Descriptive Set Theory: The Arithmetic Hierarchy 27 4 Analytic Sets 34 5 Coanalytic Sets 43 6 Determinacy 54 7 Hyperarithmetic Sets 62 II Borel Equivalence Relations 73 1 8 ¦1-Equivalence Relations 73 9 Tame Borel Equivalence Relations 82 10 Countable Borel Equivalence Relations 87 11 Hyper¯nite Equivalence Relations 92 1 These are informal notes for a course in Descriptive Set Theory given at the University of Illinois at Chicago in Fall 2002. While I hope to give a fairly broad survey of the subject we will be concentrating on problems about group actions, particularly those motivated by Vaught's conjecture. Kechris' Classical Descriptive Set Theory is the main reference for these notes. Notation: If A is a set, A<! is the set of all ¯nite sequences from A. Suppose <! σ = (a0; : : : ; am) 2 A and b 2 A. Then σ b is the sequence (a0; : : : ; am; b). We let ; denote the empty sequence. If σ 2 A<!, then jσj is the length of σ. If f : N ! A, then fjn is the sequence (f(0); : : :b; f(n ¡ 1)). If X is any set, P(X), the power set of X is the set of all subsets X. If X is a metric space, x 2 X and ² > 0, then B²(x) = fy 2 X : d(x; y) < ²g is the open ball of radius ² around x. Part I Classical Descriptive Set Theory 1 Polish Spaces De¯nition 1.1 Let X be a topological space. -
Effective Cardinals of Boldface Pointclasses
EFFECTIVE CARDINALS OF BOLDFACE POINTCLASSES ALESSANDRO ANDRETTA, GREG HJORTH, AND ITAY NEEMAN Abstract. Assuming AD + DC(R), we characterize the self-dual boldface point- classes which are strictly larger (in terms of cardinality) than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of ξ ξ Wadge rank ≤ ω1, and those of Wadge rank < ω1 when ξ is limit. 1. Introduction A boldface pointclass (for short: a pointclass) is a non-empty collection Γ of subsets of R such that Γ is closed under continuous pre-images and Γ 6= P(R). 0 0 0 Examples of pointclasses are the levels Σα, Πα, ∆α of the Borel hierarchy and the 1 1 1 levels Σn, Πn, ∆n of the projective hierarchy. In this paper we address the following Question 1. What is the cardinality of a pointclass? Assuming AC, the Axiom of Choice, Question 1 becomes trivial for all pointclasses Γ which admit a complete set. These pointclasses all have size 2ℵ0 under AC. On the other hand there is no obvious, natural way to associate, in a one-to-one way, 0 an open set (or for that matter: a closed set, or a real number) to any Σ2 set. This 0 0 suggests that in the realm of definable sets and functions already Σ1 and Σ2 may have different sizes. Indeed the second author in [Hjo98] and [Hjo02] showed that AD + V = L(R) actually implies 0 0 (a) 1 ≤ α < β < ω1 =⇒ |Σα| < |Σβ|, and 1 1 1 (b) |∆1| < |Σ1| < |Σ2| < . -
The Envelope of a Pointclass Under a Local Determinacy Hypothesis
THE ENVELOPE OF A POINTCLASS UNDER A LOCAL DETERMINACY HYPOTHESIS TREVOR M. WILSON Abstract. Given an inductive-like pointclass Γ and assuming the Ax- iom of Determinacy, Martin identified and analyzede the pointclass con- taining the norm relations of the next semiscale beyond Γ, if one exists. We show that much of Martin's analysis can be carriede out assuming only ZF + DCR + Det(∆Γ). This generalization requires arguments from Kechris{Woodin [10] ande e Martin [13]. The results of [10] and [13] can then be recovered as immediate corollaries of the general analysis. We also obtain a new proof of a theorem of Woodin on divergent models of AD+, as well as a new result regarding the derived model at an inde- structibly weakly compact limit of Woodin cardinals. Introduction Given an inductive-like pointclass Γ, Martin introduced a pointclass (which we call the envelope of Γ, following [22])e whose main feature is that it con- tains the prewellorderingse of the next scale (or semiscale) beyond Γ, if such a (semi)scale exists. This work was distributed as \Notes on the nexte Suslin cardinal," as cited in Jackson [3]. It is unpublished, so we use [3] as our reference instead for several of Martin's arguments. Martin's analysis used the assumption of the Axiom of Determinacy. We reformulate the notion of the envelope in such a way that many of its essen- tial properties can by derived without assuming AD. Instead we will work in the base theory ZF+DCR for the duration of the paper, stating determinacy hypotheses only when necessary. -
Infinitesimals
Infinitesimals: History & Application Joel A. Tropp Plan II Honors Program, WCH 4.104, The University of Texas at Austin, Austin, TX 78712 Abstract. An infinitesimal is a number whose magnitude ex- ceeds zero but somehow fails to exceed any finite, positive num- ber. Although logically problematic, infinitesimals are extremely appealing for investigating continuous phenomena. They were used extensively by mathematicians until the late 19th century, at which point they were purged because they lacked a rigorous founda- tion. In 1960, the logician Abraham Robinson revived them by constructing a number system, the hyperreals, which contains in- finitesimals and infinitely large quantities. This thesis introduces Nonstandard Analysis (NSA), the set of techniques which Robinson invented. It contains a rigorous de- velopment of the hyperreals and shows how they can be used to prove the fundamental theorems of real analysis in a direct, natural way. (Incredibly, a great deal of the presentation echoes the work of Leibniz, which was performed in the 17th century.) NSA has also extended mathematics in directions which exceed the scope of this thesis. These investigations may eventually result in fruitful discoveries. Contents Introduction: Why Infinitesimals? vi Chapter 1. Historical Background 1 1.1. Overview 1 1.2. Origins 1 1.3. Continuity 3 1.4. Eudoxus and Archimedes 5 1.5. Apply when Necessary 7 1.6. Banished 10 1.7. Regained 12 1.8. The Future 13 Chapter 2. Rigorous Infinitesimals 15 2.1. Developing Nonstandard Analysis 15 2.2. Direct Ultrapower Construction of ∗R 17 2.3. Principles of NSA 28 2.4. Working with Hyperreals 32 Chapter 3. -
Determinacy and Large Cardinals
Determinacy and Large Cardinals Itay Neeman∗ Abstract. The principle of determinacy has been crucial to the study of definable sets of real numbers. This paper surveys some of the uses of determinacy, concentrating specifically on the connection between determinacy and large cardinals, and takes this connection further, to the level of games of length ω1. Mathematics Subject Classification (2000). 03E55; 03E60; 03E45; 03E15. Keywords. Determinacy, iteration trees, large cardinals, long games, Woodin cardinals. 1. Determinacy Let ωω denote the set of infinite sequences of natural numbers. For A ⊂ ωω let Gω(A) denote the length ω game with payoff A. The format of Gω(A) is displayed in Diagram 1. Two players, denoted I and II, alternate playing natural numbers forming together a sequence x = hx(n) | n < ωi in ωω called a run of the game. The run is won by player I if x ∈ A, and otherwise the run is won by player II. I x(0) x(2) ...... II x(1) x(3) ...... Diagram 1. The game Gω(A). A game is determined if one of the players has a winning strategy. The set A is ω determined if Gω(A) is determined. For Γ ⊂ P(ω ), det(Γ) denotes the statement that all sets in Γ are determined. Using the axiom of choice, or more specifically using a wellordering of the reals, it is easy to construct a non-determined set A. det(P(ωω)) is therefore false. On the other hand it has become clear through research over the years that det(Γ) is true if all the sets in Γ are definable by some concrete means. -
The Hyperreals
THE HYPERREALS LARRY SUSANKA Abstract. In this article we define the hyperreal numbers, an ordered field containing the real numbers as well as infinitesimal numbers. These infinites- imals have magnitude smaller than that of any nonzero real number and have intuitively appealing properties, harkening back to the thoughts of the inven- tors of analysis. We use the ultrafilter construction of the hyperreal numbers which employs common properties of sets, rather than the original approach (see A. Robinson Non-Standard Analysis [5]) which used model theory. A few of the properties of the hyperreals are explored and proofs of some results from real topology and calculus are created using hyperreal arithmetic in place of the standard limit techniques. Contents The Hyperreal Numbers 1. Historical Remarks and Overview 2 2. The Construction 3 3. Vocabulary 6 4. A Collection of Exercises 7 5. Transfer 10 6. The Rearrangement and Hypertail Lemmas 14 Applications 7. Open, Closed and Boundary For Subsets of R 15 8. The Hyperreal Approach to Real Convergent Sequences 16 9. Series 18 10. More on Limits 20 11. Continuity and Uniform Continuity 22 12. Derivatives 26 13. Results Related to the Mean Value Theorem 28 14. Riemann Integral Preliminaries 32 15. The Infinitesimal Approach to Integration 36 16. An Example of Euler, Revisited 37 References 40 Index 41 Date: June 27, 2018. 1 2 LARRY SUSANKA 1. Historical Remarks and Overview The historical Euclid-derived conception of a line was as an object possessing \the quality of length without breadth" and which satisfies the various axioms of Euclid's geometric structure.