The Continuum Problem John Stillwell
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Calibrating Determinacy Strength in Borel Hierarchies
University of California Los Angeles Calibrating Determinacy Strength in Borel Hierarchies A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Sherwood Julius Hachtman 2015 c Copyright by Sherwood Julius Hachtman 2015 Abstract of the Dissertation Calibrating Determinacy Strength in Borel Hierarchies by Sherwood Julius Hachtman Doctor of Philosophy in Mathematics University of California, Los Angeles, 2015 Professor Itay Neeman, Chair We study the strength of determinacy hypotheses in levels of two hierarchies of subsets of 0 0 1 Baire space: the standard Borel hierarchy hΣαiα<ω1 , and the hierarchy of sets hΣα(Π1)iα<ω1 in the Borel σ-algebra generated by coanalytic sets. 0 We begin with Σ3, the lowest level at which the strength of determinacy had not yet been characterized in terms of a natural theory. Building on work of Philip Welch [Wel11], 0 [Wel12], we show that Σ3 determinacy is equivalent to the existence of a β-model of the 1 axiom of Π2 monotone induction. 0 For the levels Σ4 and above, we prove best-possible refinements of old bounds due to Harvey Friedman [Fri71] and Donald A. Martin [Mar85, Mar] on the strength of determinacy in terms of iterations of the Power Set axiom. We introduce a novel family of reflection 0 principles, Π1-RAPα, and prove a level-by-level equivalence between determinacy for Σ1+α+3 and existence of a wellfounded model of Π1-RAPα. For α = 0, we have the following concise 0 result: Σ4 determinacy is equivalent to the existence of an ordinal θ so that Lθ satisfies \P(!) exists, and all wellfounded trees are ranked." 0 We connect our result on Σ4 determinacy to work of Noah Schweber [Sch13] on higher order reverse mathematics. -
Arxiv:1901.02074V1 [Math.LO] 4 Jan 2019 a Xoskona Lrecrias Ih Eal Ogv Entv a Definitive a Give T to of Able Basis Be the Might Cardinals” [17]
GENERIC LARGE CARDINALS AS AXIOMS MONROE ESKEW Abstract. We argue against Foreman’s proposal to settle the continuum hy- pothesis and other classical independent questions via the adoption of generic large cardinal axioms. Shortly after proving that the set of all real numbers has a strictly larger car- dinality than the set of integers, Cantor conjectured his Continuum Hypothesis (CH): that there is no set of a size strictly in between that of the integers and the real numbers [1]. A resolution of CH was the first problem on Hilbert’s famous list presented in 1900 [19]. G¨odel made a major advance by constructing a model of the Zermelo-Frankel (ZF) axioms for set theory in which the Axiom of Choice and CH both hold, starting from a model of ZF. This showed that the axiom system ZF, if consistent on its own, could not disprove Choice, and that ZF with Choice (ZFC), a system which suffices to formalize the methods of ordinary mathematics, could not disprove CH [16]. It remained unknown at the time whether models of ZFC could be found in which CH was false, but G¨odel began to suspect that this was possible, and hence that CH could not be settled on the basis of the normal methods of mathematics. G¨odel remained hopeful, however, that new mathemati- cal axioms known as “large cardinals” might be able to give a definitive answer on CH [17]. The independence of CH from ZFC was finally solved by Cohen’s invention of the method of forcing [2]. Cohen’s method showed that ZFC could not prove CH either, and in fact could not put any kind of bound on the possible number of cardinals between the sizes of the integers and the reals. -
Georg Cantor English Version
GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle. -
Peter Koellner
Peter Koellner Department of Philosophy 50 Follen St. 320 Emerson Hall, Harvard University Cambridge, MA, 02138 Cambridge, MA 02138 (617) 821-4688 (617) 495-3970 [email protected] Employment Professor of Philosophy, Harvard University, (2010{) John L. Loeb Associate Professor of the Humanities, Harvard University, (2008{2010) Assistant Professor of Philosophy, Harvard University, (2003{2008) Education Ph.D. in Philosophy Massachusetts Institute of Technology, 2003 (Minor: Mathematical Logic) Research in Logic University of California, Berkeley Logic Group: 1998{2002 M.A. in Philosophy University of Western Ontario, 1996 B.A. in Philosophy University of Toronto, 1995 Research Interests Primary: Set Theory, Foundations of Mathematics, Philosophy of Mathematics Secondary: Foundations of Physics, Early Analytic Philosophy, History of Philosophy and the Exact Sciences Academic Awards and Honors • John Templeton Foundation Research Grant: Exploring the Frontiers of Incompleteness; 2011-2013 |$195,029 • Kurt G¨odelCentenary Research Fellowship Prize; 2008{2011 |$130,000 • John Templeton Foundation Research Grant: Exploring the Infinite (with W. Hugh Woodin); 2008{2009 |$50,000 • Clarke-Cooke Fellowships; 2005{2006, 2006{2007 • Social Sciences and Humanities Research Council of Canada Scholarship, 1996{2000 (tenure 1998{2000: Berkeley) Peter Koellner Curriculum Vitae 2 of 4 Teaching Experience Courses taught at Harvard: Set Theory (Phil 143, Fall 2003) Set Theory: Large Cardinals from Determinacy (Math 242, Fall 2004) Set Theory: Inner Model -
The Higher Infinite in Proof Theory
The Higher Infinite in Proof Theory∗y Michael Rathjen Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom 1 Introduction The higher infinite usually refers to the lofty reaches of Cantor's paradise, notably to the realm of large cardinals whose existence cannot be proved in the established for- malisation of Cantorian set theory, i.e. Zermelo-Fraenkel set theory with the axiom of choice. Proof theory, on the other hand, is commonly associated with the manipula- tion of syntactic objects, that is finite objects par excellence. However, finitary proof theory became already infinitary in the 1950's when Sch¨uttere-obtained Gentzen's ordinal analysis for number theory in a particular transparent way through the use of an infinitary proof system with the so-called !-rule (cf. [54]). Nowadays one even finds vestiges of large cardinals in ordinal-theoretic proof theory. Large cardinals have worked their way down through generalized recursion (in the shape of recursively large ordinals) to proof theory wherein they appear in the definition procedures of so-called collapsing functions which give rise to ordinal representation systems. The surprising use of ordinal representation systems employing \names" for large cardinals in current proof-theoretic ordinal analyses is the main theme of this paper. The exposition here diverges from the presentation given at the conference in two regards. Firstly, the talk began with a broad introduction, explaining the current rationale and goals of ordinal-theoretic proof theory, which take the place of the original Hilbert Program. Since this part of the talk is now incorporated in the first two sections of the BSL-paper [48] there is no point in reproducing it here. -
The Continuum Hypothesis
Mathematics As A Liberal Art Math 105 Fall 2015 Fowler 302 MWF 10:40am- 11:35am BY: 2015 Ron Buckmire http://sites.oxy.edu/ron/math/105/15/ Class 22: Monday October 26 Aleph One (@1) and All That The Continuum Hypothesis: Introducing @1 (Aleph One) We can show that the power set of the natural numbers has a cardinality greater than the the cardinality of the natural numbers. This result was proven in an 1891 paper by German mathematician Georg Cantor (1845-1918) who used something called a diagonalization argument in his proof. Using our formula for the cardinality of the power set, @0 jP(N)j = 2 = @1 The cardinality of the power set of the natural numbers is called @1. THEOREM Cantor's Power Set Theorem For any set S (finite or infinite), cardinality of the power set of S, i.e. P (S) is always strictly greater than the cardinality of S. Mathematically, this says that jP (S)j > jS THEOREM c > @0 The cardinality of the real numbers is greater than the cardinality of the natural numbers. DEFINITION: Uncountable An infinite set is said to be uncountable (or nondenumerable) if it has a cardinality that is greater than @0, i.e. it has more elements than the set of natural numbers. PROOF Let's use Cantor's diagonal process to show that the set of real numbers between 0 and 1 is uncountable. Math As A Liberal Art Class 22 Math 105 Spring 2015 THEOREM The Continuum Hypothesis, i.e. @1 =c The continuum hypothesis is that the cardinality of the continuum (i.e. -
The Continuum Hypothesis and Its Relation to the Lusin Set
THE CONTINUUM HYPOTHESIS AND ITS RELATION TO THE LUSIN SET CLIVE CHANG Abstract. In this paper, we prove that the Continuum Hypothesis is equiv- alent to the existence of a subset of R called a Lusin set and the property that every subset of R with cardinality < c is of first category. Additionally, we note an interesting consequence of the measure of a Lusin set, specifically that it has measure zero. We introduce the concepts of ordinals and cardinals, as well as discuss some basic point-set topology. Contents 1. Introduction 1 2. Ordinals 2 3. Cardinals and Countability 2 4. The Continuum Hypothesis and Aleph Numbers 2 5. The Topology on R 3 6. Meagre (First Category) and Fσ Sets 3 7. The existence of a Lusin set, assuming CH 4 8. The Lebesque Measure on R 4 9. Additional Property of the Lusin set 4 10. Lemma: CH is equivalent to R being representable as an increasing chain of countable sets 5 11. The Converse: Proof of CH from the existence of a Lusin set and a property of R 6 12. Closing Comments 6 Acknowledgments 6 References 6 1. Introduction Throughout much of the early and middle twentieth century, the Continuum Hypothesis (CH) served as one of the premier problems in the foundations of math- ematical set theory, attracting the attention of countless famous mathematicians, most notably, G¨odel,Cantor, Cohen, and Hilbert. The conjecture first advanced by Cantor in 1877, makes a claim about the relationship between the cardinality of the continuum (R) and the cardinality of the natural numbers (N), in relation to infinite set hierarchy. -
Aaboe, Asger Episodes from the Early History of Mathematics QA22 .A13 Abbott, Edwin Abbott Flatland: a Romance of Many Dimensions QA699 .A13 1953 Abbott, J
James J. Gehrig Memorial Library _________Table of Contents_______________________________________________ Section I. Cover Page..............................................i Table of Contents......................................ii Biography of James Gehrig.............................iii Section II. - Library Author’s Last Name beginning with ‘A’...................1 Author’s Last Name beginning with ‘B’...................3 Author’s Last Name beginning with ‘C’...................7 Author’s Last Name beginning with ‘D’..................10 Author’s Last Name beginning with ‘E’..................13 Author’s Last Name beginning with ‘F’..................14 Author’s Last Name beginning with ‘G’..................16 Author’s Last Name beginning with ‘H’..................18 Author’s Last Name beginning with ‘I’..................22 Author’s Last Name beginning with ‘J’..................23 Author’s Last Name beginning with ‘K’..................24 Author’s Last Name beginning with ‘L’..................27 Author’s Last Name beginning with ‘M’..................29 Author’s Last Name beginning with ‘N’..................33 Author’s Last Name beginning with ‘O’..................34 Author’s Last Name beginning with ‘P’..................35 Author’s Last Name beginning with ‘Q’..................38 Author’s Last Name beginning with ‘R’..................39 Author’s Last Name beginning with ‘S’..................41 Author’s Last Name beginning with ‘T’..................45 Author’s Last Name beginning with ‘U’..................47 Author’s Last Name beginning -
2.5. INFINITE SETS Now That We Have Covered the Basics of Elementary
2.5. INFINITE SETS Now that we have covered the basics of elementary set theory in the previous sections, we are ready to turn to infinite sets and some more advanced concepts in this area. Shortly after Georg Cantor laid out the core principles of his new theory of sets in the late 19th century, his work led him to a trove of controversial and groundbreaking results related to the cardinalities of infinite sets. We will explore some of these extraordinary findings, including Cantor’s eponymous theorem on power sets and his famous diagonal argument, both of which imply that infinite sets come in different “sizes.” We also present one of the grandest problems in all of mathematics – the Continuum Hypothesis, which posits that the cardinality of the continuum (i.e. the set of all points on a line) is equal to that of the power set of the set of natural numbers. Lastly, we conclude this section with a foray into transfinite arithmetic, an extension of the usual arithmetic with finite numbers that includes operations with so-called aleph numbers – the cardinal numbers of infinite sets. If all of this sounds rather outlandish at the moment, don’t be surprised. The properties of infinite sets can be highly counter-intuitive and you may likely be in total disbelief after encountering some of Cantor’s theorems for the first time. Cantor himself said it best: after deducing that there are just as many points on the unit interval (0,1) as there are in n-dimensional space1, he wrote to his friend and colleague Richard Dedekind: “I see it, but I don’t believe it!” The Tricky Nature of Infinity Throughout the ages, human beings have always wondered about infinity and the notion of uncountability. -
Set.1 the Power of the Continuum
set.1 The Power of the Continuum sol:set:pow: In second-order logic we can quantify over subsets of the domain, but not over explanation sec sets of subsets of the domain. To do this directly, we would need third-order logic. For instance, if we wanted to state Cantor's Theorem that there is no injective function from the power set of a set to the set itself, we might try to formulate it as \for every set X, and every set P , if P is the power set of X, then not P ⪯ X". And to say that P is the power set of X would require formalizing that the elements of P are all and only the subsets of X, so something like 8Y (P (Y ) $ Y ⊆ X). The problem lies in P (Y ): that is not a formula of second-order logic, since only terms can be arguments to one-place relation variables like P . We can, however, simulate quantification over sets of sets, if the domain is large enough. The idea is to make use of the fact that two-place relations R relates elements of the domain to elements of the domain. Given such an R, we can collect all the elements to which some x is R-related: fy 2 jMj : R(x; y)g is the set \coded by" x. Conversely, if Z ⊆ }(jMj) is some collection of subsets of jMj, and there are at least as many elements of jMj as there are sets in Z, then there is also a relation R ⊆ jMj2 such that every Y 2 Z is coded by some x using R. -
Unsolvable Problems, the Continuum Hypothesis, and the Nature of Infinity
Unsolvable problems, the Continuum Hypothesis, and the nature of infinity W. Hugh Woodin Harvard University January 9, 2017 V : The Universe of Sets The power set Suppose X is a set. The powerset of X is the set P(X ) = fY Y is a subset of X g: Cumulative Hierarchy of Sets The universe V of sets is generated by defining Vα by induction on the ordinal α: 1. V0 = ;, 2. Vα+1 = P(Vα), S 3. if α is a limit ordinal then Vα = β<α Vβ. I If X is a set then X 2 Vα for some ordinal α. I V0 = ;, V1 = f;g, V2 = f;; f;gg. I These are just the ordinals: 0, 1, and 2. I V3 has 4 elements. I This is not the ordinal 3 (in fact, it is not an ordinal). I V4 has 16 elements. I V5 has 65; 536 elements. I V1000 has a lot of elements. V! is infinite, it is the set of all (hereditarily) finite sets. The conception of V! is mathematically identical to the conception of the structure (N; +; ·). Beyond the basic axioms: large cardinal axioms The axioms I The ZFC axioms of Set Theory specify the basic axioms for V . I These axioms are naturally augmented by additional principles which assert the existence of \very large" infinite sets. I These additional principles are called large cardinal axioms. I There is a proper class of measurable cardinals. I There is a proper class of strong cardinals. I There is a proper class of Woodin cardinals. -
Forcing? Thomas Jech
WHAT IS... ? Forcing? Thomas Jech What is forcing? Forcing is a remarkably powerful case that there exists no proof of the conjecture technique for the construction of models of set and no proof of its negation. theory. It was invented in 1963 by Paul Cohen1, To make this vague discussion more precise we who used it to prove the independence of the will first elaborate on the concepts of theorem and Continuum Hypothesis. He constructed a model proof. of set theory in which the Continuum Hypothesis What are theorems and proofs? It is a use- (CH) fails, thus showing that CH is not provable ful fact that every mathematical statement can from the axioms of set theory. be expressed in the language of set theory. All What is the Continuum Hypothesis? In 1873 mathematical objects can be regarded as sets, and Georg Cantor proved that the continuum is un- relations between them can be reduced to expres- countable: that there exists no mapping of the set sions that use only the relation ∈. It is not essential N of all integers onto the set R of all real numbers. how it is done, but it can be done: For instance, Since R contains N, we have 2ℵ0 > ℵ , where 2ℵ0 0 integers are certain finite sets, rational numbers and ℵ are the cardinalities of R and N, respec- 0 are pairs of integers, real numbers are identified tively. A question arises whether 2ℵ0 is equal to with Dedekind cuts in the rationals, functions the cardinal ℵ1, the immediate successor of ℵ0.