Souslin Trees and Successors of Singular Cardinals
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Singular Cardinals: from Hausdorff's Gaps to Shelah's Pcf Theory
SINGULAR CARDINALS: FROM HAUSDORFF’S GAPS TO SHELAH’S PCF THEORY Menachem Kojman 1 PREFACE The mathematical subject of singular cardinals is young and many of the math- ematicians who made important contributions to it are still active. This makes writing a history of singular cardinals a somewhat riskier mission than writing the history of, say, Babylonian arithmetic. Yet exactly the discussions with some of the people who created the 20th century history of singular cardinals made the writing of this article fascinating. I am indebted to Moti Gitik, Ronald Jensen, Istv´an Juh´asz, Menachem Magidor and Saharon Shelah for the time and effort they spent on helping me understand the development of the subject and for many illuminations they provided. A lot of what I thought about the history of singular cardinals had to change as a result of these discussions. Special thanks are due to Istv´an Juh´asz, for his patient reading for me from the Russian text of Alexandrov and Urysohn’s Memoirs, to Salma Kuhlmann, who directed me to the definition of singular cardinals in Hausdorff’s writing, and to Stefan Geschke, who helped me with the German texts I needed to read and sometimes translate. I am also indebted to the Hausdorff project in Bonn, for publishing a beautiful annotated volume of Hausdorff’s monumental Grundz¨uge der Mengenlehre and for Springer Verlag, for rushing to me a free copy of this book; many important details about the early history of the subject were drawn from this volume. The wonderful library and archive of the Institute Mittag-Leffler are a treasure for anyone interested in mathematics at the turn of the 20th century; a particularly pleasant duty for me is to thank the institute for hosting me during my visit in September of 2009, which allowed me to verify various details in the early research literature, as well as providing me the company of many set theorists and model theorists who are interested in the subject. -
Ineffability Within the Limits of Abstraction Alone
Ineffability within the Limits of Abstraction Alone Stewart Shapiro and Gabriel Uzquiano 1 Abstraction and iteration The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. The gold standard would be a theory as rich and useful as the dominant one, ZFC, Zermelo-Fraenkel set theory with choice, perhaps aug- mented with large cardinal principles. Although the present paper is self-contained, we draw upon recent work in [32] in which we explore the power of a reasonably pure principle of reflection. To establish terminology, the Scottish neo-logicist program is to develop branches of mathematics using abstraction principles in the form: xα = xβ $ α ∼ β where the variables α and β range over items of a certain sort, x is an operator taking items of this sort to objects, and ∼ is an equivalence relation on this sort of item. The standard exemplar, of course, is Hume’s principle: #F = #G ≡ F ≈ G where, as usual, F ≈ G is an abbreviation of the second-order statement that there is a one-one correspondence from F onto G. Hume’s principle, is the main support for what is regarded as a success-story for the Scottish neo-logicist program, at least by its advocates. The search is on to develop more powerful mathematical theories on a similar basis. As witnessed by this volume and a significant portion of the contemporary liter- ature in the philosophy of mathematics, Scottish neo-logicism remains controver- sial, even for arithmetic. There are, for example, issues of Caesar and Bad Com- pany to deal with. -
Notes on Set Theory, Part 2
§11 Regular cardinals In what follows, κ , λ , µ , ν , ρ always denote cardinals. A cardinal κ is said to be regular if κ is infinite, and the union of fewer than κ sets, each of whose cardinality is less than κ , is of cardinality less than κ . In symbols: κ is regular if κ is infinite, and κ 〈 〉 ¡ κ for any set I with I ¡ < and any family Ai i∈I of sets such that Ai < ¤¦¥ £¢ κ for all i ∈ I , we have A ¡ < . i∈I i (Among finite cardinals, only 0 , 1 and 2 satisfy the displayed condition; it is not worth including these among the cardinals that we want to call "regular".) ℵ To see two examples, we know that 0 is regular: this is just to say that the union of finitely ℵ many finite sets is finite. We have also seen that 1 is regular: the meaning of this is that the union of countably many countable sets is countable. For future use, let us note the simple fact that the ordinal-least-upper-bound of any set of cardinals is a cardinal; for any set I , and cardinals λ for i∈I , lub λ is a cardinal. i i∈I i α Indeed, if α=lub λ , and β<α , then for some i∈I , β<λ . If we had β § , then by i∈I i i β λ ≤α β § λ λ < i , and Cantor-Bernstein, we would have i , contradicting the facts that i is β λ β α β ¨ α α a cardinal and < i . -
Set Theory in Computer Science a Gentle Introduction to Mathematical Modeling I
Set Theory in Computer Science A Gentle Introduction to Mathematical Modeling I Jose´ Meseguer University of Illinois at Urbana-Champaign Urbana, IL 61801, USA c Jose´ Meseguer, 2008–2010; all rights reserved. February 28, 2011 2 Contents 1 Motivation 7 2 Set Theory as an Axiomatic Theory 11 3 The Empty Set, Extensionality, and Separation 15 3.1 The Empty Set . 15 3.2 Extensionality . 15 3.3 The Failed Attempt of Comprehension . 16 3.4 Separation . 17 4 Pairing, Unions, Powersets, and Infinity 19 4.1 Pairing . 19 4.2 Unions . 21 4.3 Powersets . 24 4.4 Infinity . 26 5 Case Study: A Computable Model of Hereditarily Finite Sets 29 5.1 HF-Sets in Maude . 30 5.2 Terms, Equations, and Term Rewriting . 33 5.3 Confluence, Termination, and Sufficient Completeness . 36 5.4 A Computable Model of HF-Sets . 39 5.5 HF-Sets as a Universe for Finitary Mathematics . 43 5.6 HF-Sets with Atoms . 47 6 Relations, Functions, and Function Sets 51 6.1 Relations and Functions . 51 6.2 Formula, Assignment, and Lambda Notations . 52 6.3 Images . 54 6.4 Composing Relations and Functions . 56 6.5 Abstract Products and Disjoint Unions . 59 6.6 Relating Function Sets . 62 7 Simple and Primitive Recursion, and the Peano Axioms 65 7.1 Simple Recursion . 65 7.2 Primitive Recursion . 67 7.3 The Peano Axioms . 69 8 Case Study: The Peano Language 71 9 Binary Relations on a Set 73 9.1 Directed and Undirected Graphs . 73 9.2 Transition Systems and Automata . -
What Is Mathematics: Gödel's Theorem and Around. by Karlis
1 Version released: January 25, 2015 What is Mathematics: Gödel's Theorem and Around Hyper-textbook for students by Karlis Podnieks, Professor University of Latvia Institute of Mathematics and Computer Science An extended translation of the 2nd edition of my book "Around Gödel's theorem" published in 1992 in Russian (online copy). Diploma, 2000 Diploma, 1999 This work is licensed under a Creative Commons License and is copyrighted © 1997-2015 by me, Karlis Podnieks. This hyper-textbook contains many links to: Wikipedia, the free encyclopedia; MacTutor History of Mathematics archive of the University of St Andrews; MathWorld of Wolfram Research. Are you a platonist? Test yourself. Tuesday, August 26, 1930: Chronology of a turning point in the human intellectua l history... Visiting Gödel in Vienna... An explanation of “The Incomprehensible Effectiveness of Mathematics in the Natural Sciences" (as put by Eugene Wigner). 2 Table of Contents References..........................................................................................................4 1. Platonism, intuition and the nature of mathematics.......................................6 1.1. Platonism – the Philosophy of Working Mathematicians.......................6 1.2. Investigation of Stable Self-contained Models – the True Nature of the Mathematical Method..................................................................................15 1.3. Intuition and Axioms............................................................................20 1.4. Formal Theories....................................................................................27 -
[Math.LO] 26 Nov 2002
LOGICAL DREAMS Saharon Shelah Institute of Mathematics The Hebrew University Jerusalem, Israel Rutgers University Mathematics Department New Brunswick, NJ USA Abstract. We discuss the past and future of set theory, axiom systems and inde- pendence results. We deal in particular with cardinal arithmetic. Contents Reading instructions 0. Introductory remarks 1 What does mathematical logic do for you? 2 The glory of proven ignorance 3 Set theory and additional axioms 4 ZFC indecisiveness 5 Is ZFC really so weak? 6 Concluding remarks arXiv:math/0211398v1 [math.LO] 26 Nov 2002 2000 Mathematics Subject Classification. 03-02, 03Exx. Key words and phrases. mathematical logic, set theory, independence, incompleteness, forcing, large cardinals. I would like to thank Alice Leonhardt for the beautiful typing. This paper is based on my lecture (and the preparations to the lecture) during the conference Mathematical Challenges of the 21th Century. Publication E23 Typeset by AMS-TEX 1 2 SAHARON SHELAH Annotated Content Reading instructions. §0 Introductory remarks. [How to read; what this article tries to do; what is cardinal arithmetic.] §1 What does mathematical logic do for me? [Why is it needed; what is this animal ZFC; thesis: general results; the Jeffersonian thesis for the best framework; the scale thesis.] §2 The glory of proven ignorance. [G¨odel’s diet, Cohen’s fattening; independence in number theory; primes are random; the Riemann hypothesis, consistency strength.] §3 Set theory and additional axioms. [G¨odel sentences and large cardinals; semi-axioms; new axioms; semi-axioms on ADL[R] positive but not true; is there just one nice descriptive set theory? cardinal invariants for the continuum.] §4 ZFC indecisiveness. -
Cardinal Arithmetic: the Silver and Galvin-Hajnal Theorems
B. Zwetsloot Cardinal arithmetic: The Silver and Galvin-Hajnal Theorems Bachelor thesis 22 June 2018 Thesis supervisor: dr. K.P. Hart Leiden University Mathematical Institute Contents Introduction 1 1 Prerequisites 2 1.1 Cofinality . .2 1.2 Stationary sets . .5 2 Silver's theorem 8 3 The Galvin-Hajnal theorem 12 4 Appendix: Ordinal and cardinal numbers 17 References 21 Introduction When introduced to university-level mathematics for the first time, one of the first subjects to come up is basic set theory, as it is a necessary basis to understanding mathematics. In particular, the concept of cardinality of sets, being a measure of their size, is learned early. But what exactly are these cardinalities for objects? They turn out to be an extension of the natural numbers, originally introduced by Cantor: The cardinal numbers. These being called numbers, it is not strange to see that some standard arithmetical op- erations, like addition, multiplication and exponentiation have extensions to the cardinal numbers. The first two of these turn out to be rather uninteresting when generalized, as for infinite cardinals κ, λ we have κ + λ = κ · λ = max(κ, λ). However, exponentiation turns out to be a lot more complex, with statements like the (Generalized) Continuum Hypothesis that are independent of ZFC. As the body of results on the topic of cardinal exponentiation grew in the 60's and early 70's, set theorists became more and more convinced that except for a relatively basic in- equality, no real grip could be gained on cardinal exponentation. Jack Silver unexpectedly reversed this trend in 1974, when he showed that some cardinals, like @!1 , cannot be the first cardinal where GCH fails. -
Set-Theoretical Background 1.1 Ordinals and Cardinals
Set-Theoretical Background 11 February 2019 Our set-theoretical framework will be the Zermelo{Fraenkel axioms with the axiom of choice (ZFC): • Axiom of Extensionality. If X and Y have the same elements, then X = Y . • Axiom of Pairing. For all a and b there exists a set fa; bg that contains exactly a and b. • Axiom Schema of Separation. If P is a property (with a parameter p), then for all X and p there exists a set Y = fx 2 X : P (x; p)g that contains all those x 2 X that have the property P . • Axiom of Union. For any X there exists a set Y = S X, the union of all elements of X. • Axiom of Power Set. For any X there exists a set Y = P (X), the set of all subsets of X. • Axiom of Infinity. There exists an infinite set. • Axiom Schema of Replacement. If a class F is a function, then for any X there exists a set Y = F (X) = fF (x): x 2 Xg. • Axiom of Regularity. Every nonempty set has a minimal element for the membership relation. • Axiom of Choice. Every family of nonempty sets has a choice function. 1.1 Ordinals and cardinals A set X is well ordered if it is equipped with a total order relation such that every nonempty subset S ⊆ X has a smallest element. The statement that every set admits a well ordering is equivalent to the axiom of choice. A set X is transitive if every element of an element of X is an element of X. -
3. Cardinal Numbers
3. Cardinal Numbers Cardinality Two sets X, Y have the same cardinality (cardinal number, cardinal), (3.1) |X| = |Y |, if there exists a one-to-one mapping of X onto Y . The relation (3.1) is an equivalence relation. We assume that we can assign to each set X its cardinal number |X| so that two sets are assigned the same cardinal just in case they satisfy condition (3.1). Cardinal numbers can be defined either using the Axiom of Regularity (via equivalence classes of (3.1)), or using the Axiom of Choice. In this chapter we define cardinal numbers of well-orderable sets; as it follows from the Axiom of Choice that every set can be well-ordered, this defines cardinals in ZFC. We recall that a set X is finite if |X| = |n| for some n ∈ N;thenX is said to have n elements. Clearly, |n| = |m| if and only if n = m,andsowe define finite cardinals as natural numbers, i.e., |n| = n for all n ∈ N. The ordering of cardinal numbers is defined as follows: (3.2) |X|≤|Y | if there exists a one-to-one mapping of X into Y . We also define the strict ordering |X| < |Y | to mean that |X|≤|Y | while |X| = |Y |.Therelation≤ in (3.2) is clearly transitive. Theorem 3.2 below shows that it is indeed a par- tial ordering, and it follows from the Axiom of Choice that the ordering is linear—any two sets are comparable in this ordering. The concept of cardinality is central to the study of infinite sets. -
SCALES at ℵω in This Paper We Analyze the PCF Structure of a Generic Extension by the Main Forcing from Our Previous Paper [1
SCALES AT @! DIMA SINAPOVA AND SPENCER UNGER In this paper we analyze the PCF structure of a generic extension by the main forcing from our previous paper [19]. Our analysis relies on a different preparation than was used in the first paper. As a corollary of our analysis we have a partial answer the following question of Woodin: \Is it consistent that both the singular cardinals hypothesis and weak square fail at @!?" In particular we prove the following theorem: Theorem 0.1. It is consistent relative to a supercompact cardinal that @! @! is a strong limit, 2 = @!+2 and @!;@n fails for all n < !. The motivation for Woodin's question comes from a tension between the failure of weak square and the failure of the singular cardinals hypothesis. Definition 0.2. Let ν be a singular cardinal. The singular cardinal hypoth- esis (SCH) at ν is the assertion \If ν is strong limit, then 2ν = ν+." We note that the singular cardinals hypothesis holds at every singular car- dinal in Easton's [5] models for possible behavior of the continuum function on regular cardinals. In particular the behavior of the continuum function on singular cardinals is determined by its restriction to regular cardinals. So models for the failure of SCH give non-trivial examples of possible behavior for the continuum function on singular cardinals. The principle weak square was introduced by Jensen [9] in his study of the fine structure of L. Intermediate principles were introduced by Schimmerling [15] and it is these intermediate principles that we define here: Definition 0.3. -
Chapter 2 Constructible Sets
Chapter 2 Constructible sets In this chapter, we recall Gödel’s construction of the constructive universe L and prove that both the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH) hold in this universe. Throughout this chapter, we informally work in ZF unless otherwise stated. 2.1 The set of formulas In Sections 1.1.1 and 1.2.1, we defined the language of set theory at the meta-theoretic level. Up to now, formulas of ZF are syntactic entities that are external to set theory. However, Gödel’s construction relies on the central concept of definability (Section 2.2), that requires to manipulate formulas within the set-theoretic universe, the same way as we manipulate sets. For that, we now need to internalize the language of formulas of ZF in ZF, by constructing a particular denumerable set Form whose elements—called internal formulas or codes of formulas—are intended to represent external formulas as sets. 2.1.1 Construction of the set Form The construction of the set Form closely follows the inductive definition of the syntactic cate- gory of formulas in the meta-language (Sections 1.1.1 and 1.2.1). Variables are represented by natural numbers, letting Var = ω. When they are used as (internal) variables, natural numbers are written using the typewriter letters x, y, z, etc. (to prevent confusions with external variables, that are still written using italic letters x, y, z, etc.) For simplicity, we only consider formulas that are constructed from equality, membership, negation, disjunction and existential quantification, since all the other constructions of first- order logic can be defined from the latter using De Morgan laws. -
Constructing Cardinals from Below
Constructing Cardinals from Below W. W. Tait∗ The totality Ω of transfinite numbers was first introduced in [Cantor, 1883] by means of the principle If the initial segment Σ of Ω is a set, then it has a least strict upper bound S(Σ) ∈ Ω. Thus, for numbers α = S(Σ) and β = S(Σ0), α < β iff α ∈ Σ0; α = β iff Σ = Σ0; S(∅) is the least number 0 (although Cantor himself took the least number to be 1); if Σ has a greatest element γ, then α is its successor γ + 1; and if Σ is non-null and has no greatest element, then α is the least upper bound of Σ. The problem with the definition, of course, is in determining what it means for an initial segment to be a set. Obviously, not all of them are: for the totality Ω of all numbers is an initial segment, but to admit it as a set would yield S(Ω) < S(Ω), contradicting the assumption that Ω is well-ordered by <. Cantor himself understood this already in 1883. In his earlier writings, e.g. [1882], he had essentially defined a set ‘in some conceptual sphere’ such as arithmetic or geometry, to be the extension of a well-defined property. But in these cases, he was considering sets of objects of some type A, where being an object of type A is itself is not defined in terms of the notion of a set of objects of type A. But with his definition of the transfinite numbers, an entirely novel situation arises: the definition of Ω depends on the notion of a subset of Ω.