DOCSLIB.ORG

Preserving Levels of Projective Determinacy by Tree Forcings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Preserving Levels of Projective Determinacy by Tree Forcings Preserving levels of projective determinacy by tree forcings Fabiana Castiblancoa,∗, Philipp Schlichtb aInstitut f¨ur Mathematische Logik und Grundlagenforschung, Universit¨at M¨unster, Einsteinstraße 62, 48149 M¨unster, Germany bSchool of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, UK Abstract We prove that various classical tree forcings—for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing—preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes are added to thin projective transitive relations by these forcings. Keywords: Forcing preservation theorems, tree forcing, Projective Determinacy, thin relations, sharps, mouse operator, uniform indiscernibles 1. Motivation A. Levy and R. Solovay [28] have shown that if κ is a measurable cardinal and P is a small P 3 forcing notion, i.e. |P| < κ, then κ remains measurable in the generic extension V . Since the existence of compact, supercompact and huge cardinals, among others, are characterized by the existence of certain elementary embeddings related to ultrapowers, variants of the Levy-Solovay 6 argument were performed in that cases showing that small forcing preserves these large cardinal properties as well. Other large cardinal notions are instead characterized by the existence of extender embeddings 9 rather than simple ultrapower embeddings. In this respect, Hamkins and Woodin [17] have shown that if κ is λ-strong then it would be also λ-strong in a generic extension obtained after forcing with a small poset.1 Hence, the strongness and Woodiness of a cardinal are also preserved by 12 small forcing. Many global consequences implied by the existence of large cardinals also are preserved after forcing with certain posets. For instance, the existence of x♯ for each set of ordinals x satisfying 15 sup x ⊂ κ is a known consequence of the existence of a Ramsey cardinal κ [25, Chapter 2, §9]. Moreover, from a Ramsey cardinal κ we obtain closure under sharps for reals in the universe and 1 also, we gain Σ3-absoluteness for small generic extensions. 18 This is closely related to the following well-known result: ♯ arXiv:1810.12079v5 [math.LO] 18 Mar 2021 Fact 1.1. Suppose that for every set of ordinals x, x exists. Let P be a forcing in V and let G be P-generic over V . Then also V [G] |= ∀x (x♯ exists). ♯ 21 If we consider the property that x exists for every real x, this preservation result is no longer true. In fact, R. David (cf. [8]) has shown that in the minimal model closed under sharps for reals ♯ 1 L there is a Σ3-forcing P adding a real with no sharp in the generic extension. 24 Nevertheless, if we restrict the complexity of the forcing, we obtain positive results. Further- # more, Schlicht [36, Lemma 3.11] has proved a more general statement: given n<ω, if Mn (x) ∗Corresponding author Email addresses: [email protected] (Fabiana Castiblanco), [email protected] (Philipp Schlicht) 1In fact, they prove as well that a λ-strong cardinal κ cannot be created via any forcing of cardinality < κ (except possibly in the case of ordinals λ with cof λ ≤|P| +). Preprint submitted to Annals of Pure and Applied Logic March 19, 2021 1 exists and is ω1-iterable for every x ∈ R, then every Σ2 provably c.c.c. forcing preserves the ♯ 27 existence and iterability of Mn(x). Thus, it is natural to ask whether we can extend this result 1 to the wider class of Σ2 proper forcing notions. This paper addresses the preservation problem above and its consequences when we consider 30 Sacks (S), Silver (V), Mathias(M), Miller (ML) and Laver (L) forcing. These forcing notions are 1 proper and their complexity is at most ∆2. We prove that for each natural number n, all the ♯ ω forcing notions in the set T = {S, V, M, L, ML} preserve Mn(x) for every x ∈ ω or equivalently, 1 33 every partial order in T preserves Πn+1-determinacy (cf. Theorem 4.5). As a consequence, from ♯ 1 the existence of Mn(x) for every real x we obtain that Σn+3-P-absoluteness holds for every P ∈ T (cf. Theorem 5.4). This gives a partial answer to [21, Question 7.3]. 36 With these results in hand, we can show that given n ∈ ω, every forcing notion in T does not 1 ♯ add any new orbits to ∆n+3-thin transitive relations if we assume the existence of Mn(x) for every real x (cf. Theorem 6.10). As a motivation to this main result, we show that all the forcing notions 39 in T do not change the value of the second uniform indiscernible u2, which partially answers [21, Question 7.4]. Acknowledgements. This work is part of the Ph.D. thesis of the first author. She would like to 42 thank Ralf Schindler for his permission to include the joint results in Section 6.1 and providing background on inner model theory. The authors would further like to thank the referee for the careful reading and numerous highly useful suggestions. 45 The first author gratefully acknowledges support from the SFB 878 program “Groups, Ge- ometry & Actions” financed by the Deutsche Forschungsgemeinschaft (DFG). This project has received funding from the European Union’s Horizon 2020 research and innovation programme 48 under the Marie Sk lodowska-Curie grant agreement No 794020 (IMIC) of the second author. The second author was partially supported by FWF grant number I4039. 2. Basic notions 51 Let R denote the set of real numbers. As usual, we identify R with the power set ℘(ω) of the set ω of natural numbers, with the Baire space ωω, with the Cantor space ω2 or with the set ↑ωω of strictly increasing functions from ω to ω, depending on the context. 54 We assume familiarity of the reader with the basic facts about forcing. For undefined notions, consult the texts [22] and [37]. For our purposes, a forcing notion P consists of an underlying set P together with a preorder ≤P on P and the induced incompatibility relation ⊥P. In this case, we 57 write P = hP, ≤P, ⊥Pi. We often identify the underlying set P with P itself. 2.1. Arboreal forcing Definition 2.1. Let n ≥ 1 be a natural number. Let M be a transitive model of ZFC and let 1 1 P ∈ M be a forcing notion. We say that M is (1-step) Σn-P-absolute iff for every Σn-formula ϕ and for every real a ∈ M, we have M |= ϕ(a) ⇐⇒ M P |= ϕ(a). P V V 1 1 Π This is equivalent to the expression M ≺Σn M . Similarly, we define (1-step) n-P-absoluteness. Definition 2.2. Let n ≥ 1 be a natural number. Let M be a transitive model of ZFC. We say 1 1 Σ 1 that M is n- correct (in V ) iff M ≺Σn V . In other words, for each Σn-formula ϕ and every real a ∈ M, M |= ϕ(a) ⇐⇒ V |= ϕ(a). 1 60 In a similar way, we define Πn-correctness. 1 ω 1 Definition 2.3. Let n<ω. We say that a forcing notion P = hP, ≤P, ⊥Pi is Σn if P ⊂ ω is Σn 1 ω ω and the order and incompatibility relations ≤P and ⊥P are Σn-subsets of ω × ω. In a similar 1 1 1 1 63 way, we define Πn forcing notions. In addition, we say that P is ∆n if it is both Σn and Πn. 1 Finally, we say that P is projective if and only if P is Σn for some n<ω. 2 1 Definition 2.4. We say that a forcing notion P is Suslin if and only if it is Σ1-definable. Also, 1 66 we say that P is co-Suslin if and only if it is Π1-definable. Definition 2.5. Let P = hP, ≤P, ⊥Pi be a poset definable by a projective formula with parameter ω M M M a ∈ ω. Let M be a transitive model of ZF containing the parameter a. Then, P , P , ≤P and M ⊥P denote the forcing notion P re-interpreted in M. Also, we say that P is absolute for M if M M ≤P = ≤P ∩ M and ⊥P = ⊥P ∩ M. 1 Definition 2.6. Let n ≥ 1 be a natural number. A forcing notion P is called provably ∆n if there 1 1 is a Σn-formula ϕ and a Πn-formula ψ such that ZFC ⊢ “ϕ and ψ define the same triple P = (P, ≤P, ⊥P)” . P Definition 2.7. Let P be a forcing notion. We say that a cardinal λ is sufficiently large if λ> 2| | and we write λ ≫ P. Definition 2.8. Let P be a forcing notion and let λ ≫ P. Let M ≺ H(λ) be an elementary substructure. A condition q ∈ P is (M, P)-generic iff for every dense set D ⊂ P such that D ∈ M, D ∩ M is predense below q. Equivalently, q ∈ P is (M, P)-generic if and only if P ˙ q V “G ∩ P ∩ M is P ∩ M-generic over M”. 69 Definition 2.9. A forcing notion P is proper iff for any λ ≫ P and for any countable elementary substructure M ≺ H(λ) with P ∈ M, every p ∈ P ∩ M has an extension q ≤P p which is (M, P)- generic.
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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • ARITHMETICAL SACKS FORCING 1. Introduction Two Fundamental
    ARITHMETICAL SACKS FORCING ROD DOWNEY AND LIANG YU Abstract. We answer a question of Jockusch by constructing a hyperimmune- free minimal degree below a 1-generic one. To do this we introduce a new forcing notion called arithmetical Sacks forcing. Some other applications are presented. 1. introduction Two fundamental construction techniques in set theory and computability theory are forcing with finite strings as conditions resulting in various forms of Cohen genericity, and forcing with perfect trees, resulting in various forms of minimality. Whilst these constructions are clearly incompatible, this paper was motivated by the general question of “How can minimality and (Cohen) genericity interact?”. Jockusch [5] showed that for n ≥ 2, no n-generic degree can bound a minimal degree, and Haught [4] extended earlier work of Chong and Jockusch to show that that every nonzero Turing degree below a 1-generic degree below 00 was itself 1- generic. Thus, it seemed that these forcing notions were so incompatible that perhaps no minimal degree could even be comparable with a 1-generic one. However, this conjecture was shown to fail independently by Chong and Downey [1] and by Kumabe [7]. In each of those papers, a minimal degree below m < 00 and a 1-generic a < 000 are constructed with m < a. The specific question motivating the present paper is one of Jockusch who asked whether a hyperimmune-free (minimal) degree could be below a 1-generic one. The point here is that the construction of a hyperimmune-free degree by and large directly uses forcing with perfect trees, and is a much more “pure” form of Spector- Sacks forcing [10] and [9].
    [Show full text]
  • SET THEORY Andrea K. Dieterly a Thesis Submitted to the Graduate
    SET THEORY Andrea K. Dieterly A Thesis Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of MASTER OF ARTS August 2011 Committee: Warren Wm. McGovern, Advisor Juan Bes Rieuwert Blok i Abstract Warren Wm. McGovern, Advisor This manuscript was to show the equivalency of the Axiom of Choice, Zorn's Lemma and Zermelo's Well-Ordering Principle. Starting with a brief history of the development of set history, this work introduced the Axioms of Zermelo-Fraenkel, common applications of the axioms, and set theoretic descriptions of sets of numbers. The book, Introduction to Set Theory, by Karel Hrbacek and Thomas Jech was the primary resource with other sources providing additional background information. ii Acknowledgements I would like to thank Warren Wm. McGovern for his assistance and guidance while working and writing this thesis. I also want to thank Reiuwert Blok and Juan Bes for being on my committee. Thank you to Dan Shifflet and Nate Iverson for help with the typesetting program LATEX. A personal thank you to my husband, Don, for his love and support. iii Contents Contents . iii 1 Introduction 1 1.1 Naive Set Theory . 2 1.2 The Axiom of Choice . 4 1.3 Russell's Paradox . 5 2 Axioms of Zermelo-Fraenkel 7 2.1 First Order Logic . 7 2.2 The Axioms of Zermelo-Fraenkel . 8 2.3 The Recursive Theorem . 13 3 Development of Numbers 16 3.1 Natural Numbers and Integers . 16 3.2 Rational Numbers . 20 3.3 Real Numbers .
    [Show full text]
  • The Method of Forcing Can Be Used to Prove Theorems As Opposed to Establish Consistency Results
    THE METHOD OF FORCING JUSTIN TATCH MOORE Abstract. The purpose of this article is to give a presentation of the method of forcing aimed at someone with a minimal knowledge of set theory and logic. The emphasis will be on how the method can be used to prove theorems in ZFC. 1. Introduction Let us begin with two thought experiments. First consider the following “para- dox” in probability: if Z is a continuous random variable, then for any possible outcome z in R, the event Z 6= z occurs almost surely (i.e. with probability 1). How does one reconcile this with the fact that, in a truly random outcome, every event having probability 1 should occur? Recasting this in more formal language we have that, “for all z ∈ R, almost surely Z 6= z”, while “almost surely there exists a z ∈ R, Z = z.” Next suppose that, for some index set I, (Zi : i ∈ I) is a family of independent continuous random variables. It is a trivial matter that for each pair i 6= j, the inequality Zi 6= Zj holds with probability 1. For large index sets I, however, |{Zi : i ∈ I}| = |I| holds with probability 0; in fact this event contains no outcomes if I is larger in cardinality than R. In terms of the formal logic, we have that, “for all i 6= j in I, almost surely the event Zi 6= Zj occurs”, while “almost surely it is false that for all i 6= j ∈ I, the event Zi 6= Zj occurs”. It is natural to ask whether it is possible to revise the notion of almost surely so that its meaning remains unchanged for simple logical assertions such as Zi 6= Zj but such that it commutes with quantification.
    [Show full text]