A Beginner's Guide to Forcing
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Set Theory, by Thomas Jech, Academic Press, New York, 1978, Xii + 621 Pp., '$53.00
BOOK REVIEWS 775 BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Number 1, July 1980 © 1980 American Mathematical Society 0002-9904/80/0000-0 319/$01.75 Set theory, by Thomas Jech, Academic Press, New York, 1978, xii + 621 pp., '$53.00. "General set theory is pretty trivial stuff really" (Halmos; see [H, p. vi]). At least, with the hindsight afforded by Cantor, Zermelo, and others, it is pretty trivial to do the following. First, write down a list of axioms about sets and membership, enunciating some "obviously true" set-theoretic principles; the most popular Hst today is called ZFC (the Zermelo-Fraenkel axioms with the axiom of Choice). Next, explain how, from ZFC, one may derive all of conventional mathematics, including the general theory of transfinite cardi nals and ordinals. This "trivial" part of set theory is well covered in standard texts, such as [E] or [H]. Jech's book is an introduction to the "nontrivial" part. Now, nontrivial set theory may be roughly divided into two general areas. The first area, classical set theory, is a direct outgrowth of Cantor's work. Cantor set down the basic properties of cardinal numbers. In particular, he showed that if K is a cardinal number, then 2", or exp(/c), is a cardinal strictly larger than K (if A is a set of size K, 2* is the cardinality of the family of all subsets of A). Now starting with a cardinal K, we may form larger cardinals exp(ic), exp2(ic) = exp(exp(fc)), exp3(ic) = exp(exp2(ic)), and in fact this may be continued through the transfinite to form expa(»c) for every ordinal number a. -
A Characterization of the Canonical Extension of Boolean
A CHARACTERIZATION OF THE CANONICAL EXTENSION OF BOOLEAN HOMOMORPHISMS LUCIANO J. GONZALEZ´ Abstract. This article aims to obtain a characterization of the canon- ical extension of Boolean homomorphisms through the Stone-Cechˇ com- pactification. Then, we will show that one-to-one homomorphisms and onto homomorphisms extend to one-to-one homomorphisms and onto homomorphisms, respectively. 1. Introduction The concept of canonical extension of Boolean algebras with operators was introduced and studied by J´onsson and Tarski [13, 14]. Then, the notion of canonical extension was naturally generalized to the setting of distribu- tive lattices with operators [7]. Later on, the notion of canonical extension was generalized to the setting of lattices [8, 9, 6], and the more general setting of partially ordered sets [3]. The theory of completions for ordered algebraic structures, and in particular the theory of canonical extensions, is an important and useful tool to obtain complete relational semantics for several propositional logics such as modal logics, superintuitionistic logics, fragments of substructural logics, etc., see for instance [1, 5, 10, 16, 3]. The theory of Stone-Cechˇ compactifications of discrete spaces and in particular of discrete semigroup spaces has many applications to several branches of mathematics, see the book [12] and its references. Here we are interested in the characterization of the Stone-Cechˇ compactification of a discrete space through of the Stone duality for Boolean algebras. In this paper, we prove a characterization of the canonical extension of a arXiv:1702.06414v2 [math.LO] 1 Jun 2018 Boolean homomorphism between Boolean algebras [13] through the Stone- Cechˇ compactification of discrete spaces and the Stone duality for Boolean algebras. -
Weak Measure Extension Axioms * 1 Introduction
Weak Measure Extension Axioms Joan E Hart Union College Schenectady NY USA and Kenneth Kunen University of Wisconsin Madison WI USA hartjunionedu and kunenmathwiscedu February Abstract We consider axioms asserting that Leb esgue measure on the real line may b e extended to measure a few new nonmeasurable sets Strong versions of such axioms such as realvalued measurabilityin volve large cardinals but weak versions do not We discuss weak versions which are sucienttoprovevarious combinatorial results such as the nonexistence of Ramsey ultralters the existence of ccc spaces whose pro duct is not ccc and the existence of S and L spaces We also prove an absoluteness theorem stating that assuming our ax iom every sentence of an appropriate logical form which is forced to b e true in the random real extension of the universe is in fact already true Intro duction In this pap er a measure on a set X is a countably additive measure whose domain the measurable sets is some algebra of subsets of X The authors were supp orted by NSF Grant CCR The rst author is grateful for supp ort from the University of Wisconsin during the preparation of this pap er Both authors are grateful to the referee for a numb er of useful comments INTRODUCTION We are primarily interested in nite measures although most of our results extend to nite measures in the obvious way By the Axiom of Choice whichwealways assume there are subsets of which are not Leb esgue measurable In an attempt to measure them it is reasonable to p ostulate measure extension axioms -
Faculty Document 2436 Madison 7 October 2013
University of Wisconsin Faculty Document 2436 Madison 7 October 2013 MEMORIAL RESOLUTION OF THE FACULTY OF THE UNIVERSITY OF WISCONSIN-MADISON ON THE DEATH OF PROFESSOR EMERITA MARY ELLEN RUDIN Mary Ellen Rudin, Hilldale professor emerita of mathematics, died peacefully at home in Madison on March 18, 2013. Mary Ellen was born in Hillsboro, Texas, on December 7, 1924. She spent most of her pre-college years in Leakey, another small Texas town. In 1941, she went off to college at the University of Texas in Austin, and she met the noted topologist R.L. Moore on her first day on campus, since he was assisting in advising incoming students. He recognized her talent immediately and steered her into the math program, which she successfully completed in 1944, and she then went directly into graduate school at Austin, receiving her PhD under Moore’s supervision in 1949. After teaching at Duke University and the University of Rochester, she joined the faculty of the University of Wisconsin-Madison as a lecturer in 1959 when her husband Walter came here. Walter Rudin died on May 20, 2010. Mary Ellen became a full professor in 1971 and professor emerita in 1991. She also held two named chairs: she was appointed Grace Chisholm Young Professor in 1981 and Hilldale Professor in 1988. She received numerous honors throughout her career. She was a fellow of the American Academy of Arts and Sciences and was a member of the Hungarian Academy of Sciences, and she received honorary doctor of science degrees from the University of North Carolina, the University of the South, Kenyon College, and Cedar Crest College. -
A General Setting for the Pointwise Investigation of Determinacy
A General Setting for the Pointwise Investigation of Determinacy Yurii Khomskii? Institute of Logic, Language and Computation University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam The Netherlands Abstract. It is well-known that if we assume a large class of sets of reals to be determined then we may conclude that all sets in this class have certain regularity properties: we say that determinacy implies regularity properties classwise. In [L¨o05]the pointwise relation between determi- nacy and certain regularity properties (namely the Marczewski-Burstin algebra of arboreal forcing notions and a corresponding weak version) was examined. An open question was how this result extends to topological forcing no- tions whose natural measurability algebra is the class of sets having the Baire property. We study the relationship between the two cases, and using a definition which adequately generalizes both the Marczewski- Burstin algebra of measurability and the Baire property, prove results similar to [L¨o05]. We also show how this can be further generalized for the purpose of comparing algebras of measurability of various forcing notions. 1 Introduction The classical theorems due to Mycielski-Swierczkowski, Banach-Mazur and Mor- ton Davis respectively state that under the Axiom of Determinacy all sets of reals are Lebesgue measurable, have the Baire property and the perfect set property (see, e.g., [Ka94, pp 373{377]). In fact, these proofs give classwise implications, i.e., if Γ is a boldface pointclass (closed under continuous preimages and inter- sections with basic open sets) such that all sets in Γ are determined, then all sets in Γ have the corresponding regularity property. -
Equivalents to the Axiom of Choice and Their Uses A
EQUIVALENTS TO THE AXIOM OF CHOICE AND THEIR USES A Thesis Presented to The Faculty of the Department of Mathematics California State University, Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics By James Szufu Yang c 2015 James Szufu Yang ALL RIGHTS RESERVED ii The thesis of James Szufu Yang is approved. Mike Krebs, Ph.D. Kristin Webster, Ph.D. Michael Hoffman, Ph.D., Committee Chair Grant Fraser, Ph.D., Department Chair California State University, Los Angeles June 2015 iii ABSTRACT Equivalents to the Axiom of Choice and Their Uses By James Szufu Yang In set theory, the Axiom of Choice (AC) was formulated in 1904 by Ernst Zermelo. It is an addition to the older Zermelo-Fraenkel (ZF) set theory. We call it Zermelo-Fraenkel set theory with the Axiom of Choice and abbreviate it as ZFC. This paper starts with an introduction to the foundations of ZFC set the- ory, which includes the Zermelo-Fraenkel axioms, partially ordered sets (posets), the Cartesian product, the Axiom of Choice, and their related proofs. It then intro- duces several equivalent forms of the Axiom of Choice and proves that they are all equivalent. In the end, equivalents to the Axiom of Choice are used to prove a few fundamental theorems in set theory, linear analysis, and abstract algebra. This paper is concluded by a brief review of the work in it, followed by a few points of interest for further study in mathematics and/or set theory. iv ACKNOWLEDGMENTS Between the two department requirements to complete a master's degree in mathematics − the comprehensive exams and a thesis, I really wanted to experience doing a research and writing a serious academic paper. -
Forcing in Proof Theory∗
Forcing in proof theory¤ Jeremy Avigad November 3, 2004 Abstract Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound e®ects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation re- sults for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments. 1 Introduction In 1963, Paul Cohen introduced the method of forcing to prove the indepen- dence of both the axiom of choice and the continuum hypothesis from Zermelo- Fraenkel set theory. It was not long before Saul Kripke noted a connection be- tween forcing and his semantics for modal and intuitionistic logic, which had, in turn, appeared in a series of papers between 1959 and 1965. By 1965, Scott and Solovay had rephrased Cohen's forcing construction in terms of Boolean-valued models, foreshadowing deeper algebraic connections between forcing, Kripke se- mantics, and Grothendieck's notion of a topos of sheaves. In particular, Lawvere and Tierney were soon able to recast Cohen's original independence proofs as sheaf constructions.1 It is safe to say that these developments have had a profound impact on most branches of mathematical logic. -
Tuple Routing Strategies for Distributed Eddies
Tuple Routing Strategies for Distributed Eddies Feng Tian David J. DeWitt Department of Computer Sciences University of Wisconsin, Madison Madison, WI, 53706 {ftian, dewitt}@cs.wisc.edu Abstract data stream management systems has begun to receive the attention of the database community. Many applications that consist of streams of data Many of the fundamental assumptions that are the are inherently distributed. Since input stream basis of standard database systems no longer hold for data rates and other system parameters such as the stream management systems [8]. A typical stream query is amount of available computing resources can long running -- it listens on several continuous streams fluctuate significantly, a stream query plan must and produces a continuous stream as its result. The notion be able to adapt to these changes. Routing tuples of running time, which is used as an optimization goal by between operators of a distributed stream query a classic database optimizer, cannot be directly applied to plan is used in several data stream management a stream management system. A data stream management systems as an adaptive query optimization system must use other performance metrics. In addition, technique. The routing policy used can have a since the input stream rates and the available computing significant impact on system performance. In this resources will usually fluctuate over time, an execution paper, we use a queuing network to model a plan that works well at query installation time might be distributed stream query plan and define very inefficient just a short time later. Furthermore, the performance metrics for response time and “optimize-then-execute” paradigm of traditional database system throughput. -
Set-Theoretic Foundations
Contemporary Mathematics Volume 690, 2017 http://dx.doi.org/10.1090/conm/690/13872 Set-theoretic foundations Penelope Maddy Contents 1. Foundational uses of set theory 2. Foundational uses of category theory 3. The multiverse 4. Inconclusive conclusion References It’s more or less standard orthodoxy these days that set theory – ZFC, extended by large cardinals – provides a foundation for classical mathematics. Oddly enough, it’s less clear what ‘providing a foundation’ comes to. Still, there are those who argue strenuously that category theory would do this job better than set theory does, or even that set theory can’t do it at all, and that category theory can. There are also those who insist that set theory should be understood, not as the study of a single universe, V, purportedly described by ZFC + LCs, but as the study of a so-called ‘multiverse’ of set-theoretic universes – while retaining its foundational role. I won’t pretend to sort out all these complex and contentious matters, but I do hope to compile a few relevant observations that might help bring illumination somewhat closer to hand. 1. Foundational uses of set theory The most common characterization of set theory’s foundational role, the char- acterization found in textbooks, is illustrated in the opening sentences of Kunen’s classic book on forcing: Set theory is the foundation of mathematics. All mathematical concepts are defined in terms of the primitive notions of set and membership. In axiomatic set theory we formulate . axioms 2010 Mathematics Subject Classification. Primary 03A05; Secondary 00A30, 03Exx, 03B30, 18A15. -
Boolean-Valued Equivalence Relations and Complete Extensions of Complete Boolean Algebras
BULL. AUSTRAL. MATH. SX. MOS 02JI0, 02J05 VOL. 3 (1970), 65-72. Boolean-valued equivalence relations and complete extensions of complete boolean algebras Denis Higgs It is remarked that, if A is a complete boolean algebra and 6 is an /-valued equivalence relation on a non-empty set J , then the set of 6-extensional functions from J to A can be regarded as a complete boolean algebra extension of A and a characterization is given of the complete extensions which arise in this way. Let A be a boolean algebra, I any non-empty set. An A-valued equivalence relation on I is a function 6 : I x I •*• A such that 6(i, i) = 1 , 6(i, j) = 6(j, i) , and 6(i, j) A 6(J, k) 2 6(i, fc) for all i, j, k in J . Boolean-valued equivalence relations occur of course in boolean-valued model theory and in this context they were first introduced, so far as I know, by -Los [5], p. 103. (As it happens, it is the complement d{i, j) = 6(i, j)' of a boolean-valued equivalence relation which tos describes and he requires in addition that d{i, j) = 0 only if i = j ; such a function d(i, j) may be regarded as an /-valued metric on I . Boolean-valued metrics have been considered by a number of authors - see, for example, Ellis and Sprinkle [7], p. 25^•) Given an /J-valued equivalence relation 6 on the non-empty set I , where from now on we suppose that the boolean algebra A is complete, we Received 28 March 1970. -
FORCING and the INDEPENDENCE of the CONTINUUM HYPOTHESIS Contents 1. Preliminaries 2 1.1. Set Theory 2 1.2. Model Theory 3 2. Th
FORCING AND THE INDEPENDENCE OF THE CONTINUUM HYPOTHESIS F. CURTIS MASON Abstract. The purpose of this article is to develop the method of forcing and explain how it can be used to produce independence results. We first remind the reader of some basic set theory and model theory, which will then allow us to develop the logical groundwork needed in order to ensure that forcing constructions can in fact provide proper independence results. Next, we develop the basics of forcing, in particular detailing the construction of generic extensions of models of ZFC and proving that those extensions are themselves models of ZFC. Finally, we use the forcing notions Cκ and Kα to prove that the Continuum Hypothesis is independent from ZFC. Contents 1. Preliminaries 2 1.1. Set Theory 2 1.2. Model Theory 3 2. The Logical Justification for Forcing 4 2.1. The Reflection Principle 4 2.2. The Mostowski Collapsing Theorem and Countable Transitive Models 7 2.3. Basic Absoluteness Results 8 3. The Logical Structure of the Argument 9 4. Forcing Notions 9 5. Generic Extensions 12 6. The Forcing Relation 14 7. The Independence of the Continuum Hypothesis 18 Acknowledgements 23 References 23 The Continuum Hypothesis (CH) is the assertion that there are no cardinalities strictly in between that of the natural numbers and that of the reals, or more for- @0 mally, 2 = @1. In 1940, Kurt G¨odelshowed that both the Axiom of Choice and the Continuum Hypothesis are relatively consistent with the axioms of ZF ; he did this by constructing a so-called inner model L of the universe of sets V such that (L; 2) is a (class-sized) model of ZFC + CH. -
Higher Randomness and Forcing with Closed Sets
Higher randomness and forcing with closed sets Benoit Monin Université Paris Diderot, LIAFA, Paris, France [email protected] Abstract 1 Kechris showed in [8] that there exists a largest Π1 set of measure 0. An explicit construction of 1 this largest Π1 nullset has later been given in [6]. Due to its universal nature, it was conjectured by many that this nullset has a high Borel rank (the question is explicitely mentioned in [3] and 0 [16]). In this paper, we refute this conjecture and show that this nullset is merely Σ3. Together 0 with a result of Liang Yu, our result also implies that the exact Borel complexity of this set is Σ3. To do this proof, we develop the machinery of effective randomness and effective Solovay gen- ericity, investigating the connections between those notions and effective domination properties. 1998 ACM Subject Classification F.4.1 Mathematical Logic Keywords and phrases Effective descriptive set theory, Higher computability, Effective random- ness, Genericity Digital Object Identifier 10.4230/LIPIcs.STACS.2014.566 1 Introduction We will study in this paper the notion of forcing with closed sets of positive measure and several variants of it. This forcing is generally attributed to Solovay, who used it in [15] to produce a model of ZF +DC in which all sets of reals are Lebesgue measurable. Stronger and stronger genericity for this forcing coincides with stronger and stronger notions of randomness. It is actually possible to express most of the randomness definitions that have been made over the years by forcing over closed sets of positive measure.