What Is Forcing Anyway?
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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. -
UNIVERSALLY BAIRE SETS and GENERIC ABSOLUTENESS TREVOR M. WILSON Introduction Generic Absoluteness Principles Assert That Certai
UNIVERSALLY BAIRE SETS AND GENERIC ABSOLUTENESS TREVOR M. WILSON Abstract. We prove several equivalences and relative consistency re- 2 uBλ sults involving notions of generic absoluteness beyond Woodin's (Σ1) generic absoluteness for a limit of Woodin cardinals λ. In particular,e we R 2 uBλ prove that two-step 9 (Π1) generic absoluteness below a measur- able cardinal that is a limite of Woodin cardinals has high consistency 2 uBλ strength, and that it is equivalent with the existence of trees for (Π1) formulas. The construction of these trees uses a general method for building an absolute complement for a given tree T assuming many \failures of covering" for the models L(T;Vα) below a measurable car- dinal. Introduction Generic absoluteness principles assert that certain properties of the set- theoretic universe cannot be changed by the method of forcing. Some pro- perties, such as the truth or falsity of the Continuum Hypothesis, can always be changed by forcing. Accordingly, one approach to formulating generic ab- soluteness principles is to consider properties of a limited complexity such 1 1 as those corresponding to pointclasses in descriptive set theory: Σ2, Σ3, projective, and so on. (Another approach is to limit the class ofe allowede forcing notions. For a survey of results in this area, see [1].) Shoenfield’s 1 absoluteness theorem implies that Σ2 statements are always generically ab- solute. Generic absoluteness principlese for larger pointclasses tend to be equiconsistent with strong axioms of infinity, and they may also relate to the extent of the universally Baire sets. 1 For example, one-step Σ3 generic absoluteness is shown in [3] to be equiconsistent with the existencee of a Σ2-reflecting cardinal and to be equiv- 1 alent with the statement that every ∆2 set of reals is universally Baire. -
E-RECURSIVE INTUITIONS in Memoriam Professor Joseph R. Shoenfield Contents 1. Initial Intuitions 1 2. E-Recursively Enumerable V
E-RECURSIVE INTUITIONS GERALD E. SACKS In Memoriam Professor Joseph R. Shoenfield Abstract. An informal sketch (with intermittent details) of parts of E-Recursion theory, mostly old, some new, that stresses intu- ition. The lack of effective unbounded search is balanced by the availability of divergence witnesses. A set is E-closed iff it is tran- sitive and closed under the application of partial E-recursive func- tions. Some finite injury, forcing, and model theoretic construc- tions can be adapted to E-closed sets that are not 1 admissible. Reflection plays a central role. Contents 1. Initial Intuitions 1 2. E-Recursively Enumerable Versus 1 4 3. E-Recursion Versus -Recursion 6 4. Selection and Reflection 8 5. Finite Injury Arguments and Post’sProblem 12 6. Forcing: C.C.C. Versus Countably Closed 13 7. Model-Theoretic Completeness and Compactness 14 References 16 1. Initial Intuitions One of the central intuitions of classical recursion theory is the effec- tiveness of unbounded search. Let A be a nonempty recursively enu- merable set of nonnegative integers. A member of A can be selected by simply enumerating A until some member appears. This procedure, known as unbounded search, consists of following instructions until a termination point is reached. What eventually appears is not merely Date: December 10, 2010. My thanks to the organizers of "Effective Mathematics of the Uncountable," CUNY Graduate Center, August 2009, for their kind invitation. 1 E-RECURSIVEINTUITIONS 2 some number n A but a computation that reveals n A. Unbounded search in its full2 glory consists of enumerating all computations2 until a suitable computation, if it exists, is found. -
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. -
Strong Logics of First and Second Order∗
Strong Logics of First and Second Order∗ Peter Koellner Abstract In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin’s Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine’s claim that second-order logic is really set theory in sheep’s clothing. This paper is concerned with strong logics of first and second order. At the most abstract level, a strong logic of first-order has the following general form: Let L be a first-order language and let Φ(x) be a formula that defines a class of L-structures. Then, for a recursively enumerable set T of sentences of L, and for a sentence ϕ of L set T |=Φ(x) ϕ iff for all L-structures M such that Φ(M), if M |= T then M |= ϕ. -
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. -
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. -
Determinacy from Strong Reflection
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 8, August 2014, Pages 4443–4490 S 0002-9947(2013)06058-8 Article electronically published on May 28, 2013 DETERMINACY FROM STRONG REFLECTION JOHN STEEL AND STUART ZOBLE Abstract. The Axiom of Determinacy holds in the inner model L(R) assum- ing Martin’s Maximum for partial orderings of size c. 1. Introduction A theorem of Neeman gives a particularly elegant sufficient condition for a set B of reals to be determined: B is determined if there is a triple (M,τ,Σ) which captures B in the sense that M is a model of a sufficient fragment of set theory, τ is a forcing term in M with respect to the collapse of some Woodin cardinal δ of M to be countable, and Σ is an ω + 1-iteration strategy for M such that B ∩ N[g]=i(τ)g, whenever i : M → N is an iteration map by Σ, and g is generic over N for the collapse of i(δ). The core model induction, the subject of the forthcoming book [16], is a method pioneered by Woodin for constructing such triples (M,τ,Σ) by induction on the complexity of the set B. It seems to be the only generally applicable method for making fine consistency strength calculations above the level of one Woodin cardinal. We employ this method here to establish that the Axiom of Determinacy holds in the inner model L(R) from consequences of the maximal forcing axiom MM(c), or Martin’s Maximum for partial orderings of size c.