DOCSLIB.ORG

TRANSFINITE DEPENDENT CHOICE and Co-MODEL REFLECTION §1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
TRANSFINITE DEPENDENT CHOICE and Co-MODEL REFLECTION §1 THE JOURNAL OF SYMBOLIC Louie Volume 67. Number 3, Sept. 2002 TRANSFINITE DEPENDENT CHOICE AND co-MODEL REFLECTION CHRISTIAN RUEDE Abstract. In this paper we present some metapredicative subsystems of analysis. We deal with reflection principles, w-model existence axioms (limit axioms) and axioms asserting the existence of hierarchies. We show several equivalences among the introduced subsystems. In particular we prove the equivalence of l] ] transfinite dependent choice and Tl 2 reflection on cu-models of LJ-DC. §1. Introduction. The formal system of classical analysis is second order arith­ metic with full comprehension principle. It was called classical analysis, since clas­ sical mathematical analysis can be formalized in it. Often, subsystems of classical analysis suffice as formal framework for particular parts of mathematical analysis. During the last decades a lot of such subsystems have been isolated and proof- theoretically investigated. The subsystems of analysis introduced in this paper be­ long to metapredicative proof-theory. Metapredicative systems have proof-theoretic ordinals beyond To but can still be treated by methods of predicative proof-theory only. Recently, numerous interesting metapredicative systems have been character­ ized. For previous work in metapredicativity the reader is referred to Jager [3], Jager, Kahle, Setzer and Strahm [4], Jager and Strahm [5, 6], Kahle [7], Rathjen [8] andStrahm[ll, 12, 13]. Metapredicative subsystems of analysis are for instance: ATR (proof-theoretic ordinal r£0, e.g., [5]), ATR + Sj-DC, ATR0 + Ej-DC (proof-theoretic ordinal cple00, (plcoO respectively, [5]) and FTR, FTR0 (proof-theoretic ordinal ip20eo, ip200 re­ spectively, [12]). We introduce in this paper a lot of subsystems of analysis with proof-theoretic ordinals between <^200 and tpsoOO. Three concepts are of central importance in this paper: co-models, reflections and hierarchies. Each subsystem, which we shall introduce, deals with one of these concepts. We shall prove equivalences of some subsystems and determine the proof- theoretic ordinal of some of them. To prove these equivalences, we use the method of "pseudohierarchies" (cf. [10]). In order to define co-models within subsystems of analysis we have to formalize the notion of an co-model. This leads to the notion of countable coded co-model, cf. e.g., [10]. We say that M satisfies <p or that M is an co-model of tp iff M M reflects tp, i.e., iff <p holds. For instance, if ^4XACA is a finite axiomatization of (ACA), then M is an co-model of ACA iff (AX^CA)™ holds. In the following we Received September 30, 2000; revised January 20, 2002. This paper comprises a part of the author's Ph.D. dissertation [9]. © 2002, Association for Symbolic Logic 0022-4812/02/6703-0019/S2.60 1153 Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 08:50:19, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.2178/jsl/1190150155 1154 CHRISTIAN RUEDE shall assert the existence of co-models by so-called limit axioms. They are of the M form (\/X)(3M)(X G M A ip ) for a given ip. For instance, if ^xEi.AC is a finite axiomatization of (ACA) + (E{-AC), then the axiom M (1) (VX)(3M)(X € M A UxE].AC) ) says that for each set X there exists an co -model of 2 J -AC containing X. In a certain sense a limit axiom is a special kind of a reflection principle. For any given formula the limit axiom asserts the existence of a set M reflecting this formula. On the other hand, a reflection principle asserts that for each formula <p belonging to a given class of formulas, there exists a set M reflecting the formula ip. For instance, the axiom scheme (IIj-RFN) is of this type. (II^-RFN) For all 5?t formulas <p[x, Z] in 1^: <p[x, Z] -• (3X)(Z G X A (AXACAV A tpx[x, Z]). We shall now discuss hierarchies. Fix a well-ordering Z. By a hierarchy along Z we mean a set Y, such that for each a infield(Z) we have <p{Yza, Ya) for a given formula <p. YZa is the disjoint union of all stages Yb such that b is Z-less than a. The formula y> characterizes the hierarchy and typically for b Z-less than a Ya is in some sense more complex and contains more information than Yh. A typical example is the hyperarithmetical hierarchy. We discuss two kinds of axioms which assure the existence of hierarchies. On the one hand we have axioms which claim directly the existence of the hierarchy. For instance (ATR) and (FTR) are of this form. (ATR) For all arithmetic 2?2 formulas ip(x, X): WO{Z) -> (3F)(Va &field{Z))(Vx)(x G Ya <-> <p(x, YZa)). As a second group we discuss axioms of the form "If we can build steps, then the corresponding stairs exist". For example (Ej-DC) is of this form. The resulting hierarchies are along co. (FTR) For all ^-positive, arithmetic 2?i formulas tp(X, Y,x, a): WO(Z) -» (3Z)(Va efield{Z))(\fx){x G Xa <-• ip(Xa, XZa,x,a)). The main questions addressed in this paper are 1. Are there equivalences among limit axioms, reflection principles and hierarchy existence axioms? For instance, what co-models can be built given the existence of certain hierarchies and vice versa? 2. Are there further natural metapredicative subsystems of analysis? An example of the equivalence described in point 1 is the equivalence of (Ej-DC) and (n^-RFN) over ACA0 (cf. [10, theorem VIII.5.12]) or the equivalence of (ATR) and (1) (cf. [10, theorem VIII.4.20] — the difficult direction is shown there). We prove in this paper a lot of other such equivalences and we shall see that point 1 and point 2 mutually elucidate each other. In particular we shall prove the following equivalences over ACAo. I. Ej transfinite dependent choice is equivalent to X\\ reflection on co-models of Ej-DC. II. Existence of fixed point hierarchies is equivalent to existence of co-models of ATR + E} -DC and to Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 08:50:19, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.2178/jsl/1190150155 TRANSF1NITE DEPENDENT CHOICE AND co-MODEL REFLECTION 1155 existence of hierarchies of co -models of £ j -AC and to existence of hierarchies of co-models of ATR. Since the proof-theoretic ordinal of FTRo is ^200 (cf. [12]), the corresponding theories to the axioms listed in II have proof-theoretic ordinal <^200, too. In [9] it is proved that ipcoOO is the proof-theoretic ordinal of the theories corresponding to the axioms listed in I1. Moreover, the proof-theoretic analysis of these theories uses methods of predicative proof-theory only. Hence they are metapredicative. §2. Preliminaries. In this section we fix notations and abbreviations and intro­ duce some well-known subsystems of analysis. We let S?2 denote the language of second order arithmetic. 2^2 includes number variables (denoted by small letters, except r,s,t), set variables (denoted by capital letters), symbols for all primitive recursive functions and relations, the symbol e for elementhood between numbers and sets, as well as equality in the first sort. The number terms r, s, t of Si and the formulas ip,y/,6,... of Sfi are defined as usual. An JS?2 formula is called arithmetic, if it does not contain bound set variables (but possibly free set variables). For the collection of these formulas we write Flo. Y\ is the collection of all arithmetic formulas and of all .S?2 formulas 3X<p(X) with ip(X) from IIQ. Analogously £{ and 11} are defined. In the following (...) denotes a primitive recursive coding function for n -tuples (t\,..., t„) with associated projections (-)i, • • •, (•)«• Seq„ is the primitive recursive set of sequence numbers of length n. We write X fox X\,...,X„,s e X, for (s, t) e X and r £ Xt:S for {{r, s), t) e X. Occasionally we use the following abbreviations. X = Y = (Vx)(x e X <-• x e Y), X ^ Y = <X=Y), x e Y = (3k)(\/x){x £IH (x,k) e Y), OYez)cp(Y) = (3k)<p(Yk), (vrezV(r) = (Vk)<p{Yk), X e Y = Xx e Y A • • • A X„ <= Y wo(x) = formalization of "X codes a (non-reflexive) well-ordering", x efield(X) = (3y)({x,y)eXV(y,x)GX), x € YZa = Seqix Ax £ Y A ((x)\,a) G Z. YZa is the disjoint union of all projections Yb with {b, a) € Z. For a well-ordering Z we let 0z denote the Z-least element in field(Z) and for a efield{Z) we let a +z 1 denote the Z-least element infield(Z) Z-greater than a, when it exists. Sometimes we write aZb for (a, b) £ Z. Furthermore, we write <p[x, X] if all free number variables of ip are among x and all free set variables of tp are among X. We write <p[x\f, X\ Y] for the formula ip 1 In this paper we are chiefly interested in the proofs of the equivalences listed in I and II and not in the proof-theoretic analysis of the corresponding theories. Hence we do not define the terms "proof-theoretic ordinal" and "notation system", but refer to [4. 6, 9], Downloaded from https:/www.cambridge.org/core.
Load more

Recommended publications
  • How Many Borel Sets Are There? Object. This Series of Exercises Is
    How Many Borel Sets are There? Object. This series of exercises is designed to lead to the conclusion that if BR is the σ- algebra of Borel sets in R, then Card(BR) = c := Card(R): This is the conclusion of problem 4. As a bonus, we also get some insight into the \structure" of BR via problem 2. This just scratches the surface. If you still have an itch after all this, you want to talk to a set theorist. This treatment is based on the discussion surrounding [1, Proposition 1.23] and [2, Chap. V x10 #31]. For these problems, you will need to know a bit about well-ordered sets and transfinite induction. I suggest [1, x0.4] where transfinite induction is [1, Proposition 0.15]. Note that by [1, Proposition 0.18], there is an uncountable well ordered set Ω such that for all x 2 Ω, Ix := f y 2 Ω: y < x g is countable. The elements of Ω are called the countable ordinals. We let 1 := inf Ω. If x 2 Ω, then x + 1 := inff y 2 Ω: y > x g is called the immediate successor of x. If there is a z 2 Ω such that z + 1 = x, then z is called the immediate predecessor of x. If x has no immediate predecessor, then x is called a limit ordinal.1 1. Show that Card(Ω) ≤ c. (This follows from [1, Propositions 0.17 and 0.18]. Alternatively, you can use transfinite induction to construct an injective function f :Ω ! R.)2 ANS: Actually, this follows almost immediately from Folland's Proposition 0.17.
    [Show full text]
  • Epsilon Substitution for Transfinite Induction
    Epsilon Substitution for Transfinite Induction Henry Towsner May 20, 2005 Abstract We apply Mints’ technique for proving the termination of the epsilon substitution method via cut-elimination to the system of Peano Arith- metic with Transfinite Induction given by Arai. 1 Introduction Hilbert introduced the epsilon calculus in [Hilbert, 1970] as a method for proving the consistency of arithmetic and analysis. In place of the usual quantifiers, a symbol is added, allowing terms of the form xφ[x], which are interpreted as “some x such that φ[x] holds, if such a number exists.” When there is no x satisfying φ[x], we allow xφ[x] to take an arbitrary value (usually 0). Then the existential quantifier can be defined by ∃xφ[x] ⇔ φ[xφ[x]] and the universal quantifier by ∀xφ[x] ⇔ φ[x¬φ[x]] Hilbert proposed a method for transforming non-finitistic proofs in this ep- silon calculus into finitistic proofs by assigning numerical values to all the epsilon terms, making the proof entirely combinatorial. The difficulty centers on the critical formulas, axioms of the form φ[t] → φ[xφ[x]] In Hilbert’s method we consider the (finite) list of critical formulas which appear in a proof. Then we define a series of functions, called -substitutions, each providing values for some of the of the epsilon terms appearing in the list. Each of these substitutions will satisfy some, but not necessarily all, of the critical formulas appearing in proof. At each step we take the simplest unsatisfied critical formula and update it so that it becomes true.
    [Show full text]
  • Elements of Set Theory
    Elements of set theory April 1, 2014 ii Contents 1 Zermelo{Fraenkel axiomatization 1 1.1 Historical context . 1 1.2 The language of the theory . 3 1.3 The most basic axioms . 4 1.4 Axiom of Infinity . 4 1.5 Axiom schema of Comprehension . 5 1.6 Functions . 6 1.7 Axiom of Choice . 7 1.8 Axiom schema of Replacement . 9 1.9 Axiom of Regularity . 9 2 Basic notions 11 2.1 Transitive sets . 11 2.2 Von Neumann's natural numbers . 11 2.3 Finite and infinite sets . 15 2.4 Cardinality . 17 2.5 Countable and uncountable sets . 19 3 Ordinals 21 3.1 Basic definitions . 21 3.2 Transfinite induction and recursion . 25 3.3 Applications with choice . 26 3.4 Applications without choice . 29 3.5 Cardinal numbers . 31 4 Descriptive set theory 35 4.1 Rational and real numbers . 35 4.2 Topological spaces . 37 4.3 Polish spaces . 39 4.4 Borel sets . 43 4.5 Analytic sets . 46 4.6 Lebesgue's mistake . 48 iii iv CONTENTS 5 Formal logic 51 5.1 Propositional logic . 51 5.1.1 Propositional logic: syntax . 51 5.1.2 Propositional logic: semantics . 52 5.1.3 Propositional logic: completeness . 53 5.2 First order logic . 56 5.2.1 First order logic: syntax . 56 5.2.2 First order logic: semantics . 59 5.2.3 Completeness theorem . 60 6 Model theory 67 6.1 Basic notions . 67 6.2 Ultraproducts and nonstandard analysis . 68 6.3 Quantifier elimination and the real closed fields .
    [Show full text]
  • Ordinals. Transfinite Induction and Recursion We Will Not Use The
    Set Theory (MATH 6730) Ordinals. Transfinite Induction and Recursion We will not use the Axiom of Choice in this chapter. All theorems we prove will be theorems of ZF = ZFC n fACg. 1. Ordinals To motivate ordinals, recall from elementary (naive) set theory that natural numbers can be used to label elements of a finite set to indicate a linear ordering: 0th, 1st, 2nd, :::, 2019th element. Although this will not be immediately clear from the definition, `ordinal numbers' or `ordi- nals' are sets which are introduced to play the same role for special linear orderings, called `well-orderings', of arbitrary sets. For example, one such ordering might be the following: 0th, 1st, 2nd, :::, 2019th, :::;!th, (! + 1)st element: | {z } we have run out of all natural numbers Definition 1.1. A set s is called transitive, if every element of s is a subset of s; i.e., for any sets x; y such that x 2 y 2 s we have that x 2 s. Thus, the class of all transitive sets is defined by the formula tr(s) ≡ 8x 8y (x 2 y ^ y 2 s) ! x 2 s: Definition 1.2. A transitive set of transitive sets is called an ordinal number or ordinal. Thus, the class On of all ordinals is defined by the formula on(x) ≡ tr(x) ^ 8y 2 x tr(y): Ordinals will usually be denoted by the first few letters of the Greek alphabet. The following facts are easy consequences of the definition of ordinals. 1 2 Theorem 1.3. (i) ; 2 On.
    [Show full text]
  • Towards a Probability-Free Theory of Continuous Martingales
    Towards a probability-free theory of continuous martingales Vladimir Vovk and Glenn Shafer volodya.vovk,glennrayshafer @gmail.com { } March 28, 2017 Abstract Without probability theory, we define classes of supermartingales, martingales, and semimartingales in idealized financial markets with con- tinuous price paths. This allows us to establish probability-free versions of a number of standard results in martingale theory, including the Dubins– Schwarz theorem, the Girsanov theorem, and results concerning the Itˆo integral. We also establish the existence of an equity premium and a CAPM relationship in this probability-free setting. The version of this paper at http://probabilityandfinance.com (Working Paper 45) is updated most often. 1 Introduction We consider a financial market in which a finite number of securities with con- tinuous price paths are traded. We do not make any stochastic assumptions, and our basic definitions are in the spirit of what we call game-theoretic prob- ability (see, e.g., [9]). This theory’s key notion, probability-free superhedging, has been formalized in different ways in the case of continuous time. The def- inition suggested in [10] is very cautious. Perkowski and Pr¨omel [5] propose a broader definition, making superhedging easier. In this paper we propose an arXiv:1703.08715v1 [q-fin.MF] 25 Mar 2017 even broader definition of superhedging and use it to define concepts such as continuous martingale, nonnegative supermartingale, etc. This allows us to de- rive probability-free versions of several standard results of martingale theory. In particular, we will give a simple proof of the Girsanov theorem stated (and proved in a roundabout way) in [13].
    [Show full text]
  • Some Set Theory We Should Know Cardinality and Cardinal Numbers
    SOME SET THEORY WE SHOULD KNOW CARDINALITY AND CARDINAL NUMBERS De¯nition. Two sets A and B are said to have the same cardinality, and we write jAj = jBj, if there exists a one-to-one onto function f : A ! B. We also say jAj · jBj if there exists a one-to-one (but not necessarily onto) function f : A ! B. Then the SchrÄoder-BernsteinTheorem says: jAj · jBj and jBj · jAj implies jAj = jBj: SchrÄoder-BernsteinTheorem. If there are one-to-one maps f : A ! B and g : B ! A, then jAj = jBj. A set is called countable if it is either ¯nite or has the same cardinality as the set N of positive integers. Theorem ST1. (a) A countable union of countable sets is countable; (b) If A1;A2; :::; An are countable, so is ¦i·nAi; (c) If A is countable, so is the set of all ¯nite subsets of A, as well as the set of all ¯nite sequences of elements of A; (d) The set Q of all rational numbers is countable. Theorem ST2. The following sets have the same cardinality as the set R of real numbers: (a) The set P(N) of all subsets of the natural numbers N; (b) The set of all functions f : N ! f0; 1g; (c) The set of all in¯nite sequences of 0's and 1's; (d) The set of all in¯nite sequences of real numbers. The cardinality of N (and any countable in¯nite set) is denoted by @0. @1 denotes the next in¯nite cardinal, @2 the next, etc.
    [Show full text]
  • Transfinite Induction for Measure Theory (Corrected Aug 30, 2013)
    LECTURE NOTES: TRANSFINITE INDUCTION FOR MEASURE THEORY (CORRECTED AUG 30, 2013) MATH 671 FALL 2013 (PROF: DAVID ROSS, DEPARTMENT OF MATHEMATICS) Start with a quick discussion of cardinal and ordinal numbers. Cardinals are a measure of size, ordinals of ordering. Ideally you should find an real introduction to this somewhere, either in a Math 454 text or in a mid-20th-century analysis text. 1. Cardinality Definition 1.1. A is equinumerous with B provided there is a bijection from A onto B. Other, equivalent notation/terminology for \A is equinumerous with B": (1) A and B have the same cardinality (2) A ≈ B (3) card(A) = card(B) We seem to be referring to a concept called \cardinality" without really defining it. Technically, we should view all the above as simply complicated ways of express- ing a relationship between the two sets A and B. Early treatments of set theory tried to define the cardinality of a set to be its equivalence class under the relation ≈, but since ≈ is defined on the class of all sets (which is not itself a set), this leads to technical problems related to the classical set-theoretic paradoxes. Nevertheless, it is not a terrible way to think of cardinalities. Some important cardinalities: •@0 =cardinality of N. Sets with this cardinality (ie, which can be put into a 1-1 correspondence with N) are called countably infinite. • 2@0 = c =cardinality of R (the continuum). Any math grad student should be able to prove that this is also the cardinality of P(N). •@1 =the first uncountable cardinal.
    [Show full text]
  • Transfinite Induction
    Transfinite Induction G. Eric Moorhouse, University of Wyoming Notes for ACNT Seminar, 20 Jan 2009 Abstract Let X ⊂ R3 be the complement of a single point. We prove, by transfinite induction, that X can be partitioned into lines. This result is intended as an introduction to transfinite induction. 1 Cardinality Let A and B be sets. We write |A| = |B| (in words, A and B have the same cardinality) if there exists a bijection A → B; |A| à |B| if there exists an injection (a one-to-one map) A → B; |A| < |B| if |A| à |B| but there is no bijection from A to B. The Cantor-Bernstein-Schroeder Theorem asserts that if |A| à |B| and |B| à |A|, then |A| = |B|. (The proof is elementary but requires some thought.) So we may reasonably speak of the cardinality of a set, as a measure of its size, and compare any two sets according to their cardinalities. The cardinality of Z is denoted ℵ0. Every set A having |A| à ℵ0 is called countable; in this case either A is finite, or |A|=ℵ0 (countably infinite). Examples of countably infinite sets include {positive integers},2Z ={even integers}, and Q. Cantor showed that R is uncountable; we denote |R| = 2ℵ0. This cardinality, which we call the cardinality of the continuum, strictly exceeds ℵ0; it is also the cardinality of Rn for every positive integer n. In this context it is almost obligatory to mention the Continuum Hypothe- ℵ sis (CH), which is the assertion that there is no set A satisfying ℵ0 < |A| < 2 0.
    [Show full text]
  • On the Necessary Use of Abstract Set Theory
    ADVANCES IN MATHEMATICS 41, 209-280 (1981) On the Necessary Use of Abstract Set Theory HARVEY FRIEDMAN* Department of Mathematics, Ohio State University, Columbus, Ohio 43210 In this paper we present some independence results from the Zermelo-Frankel axioms of set theory with the axiom of choice (ZFC) which differ from earlier such independence results in three major respects. Firstly, these new propositions that are shown to be independent of ZFC (i.e., neither provable nor refutable from ZFC) form mathematically natural assertions about Bore1 functions of several variables from the Hilbert cube I” into the unit interval, or back into the Hilbert cube. As such, they are of a level of abstraction significantly below that of the earlier independence results. Secondly, these propositions are not only independent of ZFC, but also of ZFC together with the axiom of constructibility (V = L). The only earlier examples of intelligible statements independent of ZFC + V= L either express properties of formal systems such as ZFC (e.g., the consistency of ZFC), or assert the existence of very large cardinalities (e.g., inaccessible cardinals). The great bulk of independence results from ZFCLthe ones that involve standard mathematical concepts and constructions-are about sets of limited cardinality (most commonly, that of at most the continuum), and are obtained using the forcing method introduced by Paul J. Cohen (see [2]). It is now known in virtually every such case, that these independence results are eliminated if V= L is added to ZFC. Finally, some of our propositions can be proved in the theory of classes, as formalized by the Morse-Kelley class theory with the axiom of choice for sets (MKC), but not in ZFC.
    [Show full text]
  • Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions
    This is a repository copy of Constructive zermelo-fraenkel set theory, power set, and the calculus of constructions. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/75182/ Book Section: Rathjen, M (2012) Constructive zermelo-fraenkel set theory, power set, and the calculus of constructions. In: Dybjer, P, Lindström, S, Palmgren, E and Sundholm, G, (eds.) Epistemology versus ontology: Essays on the philosophy and foundations of mathematics in honour of Per Martin-Löf. Logic, Epistemology, and the Unity of Science, 27 . Springer , Dordrecht, Netherlands , 313 - 349. ISBN 9789400744356 https://doi.org/10.1007/978-94-007-4435-6 Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions Michael Rathjen∗ Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom [email protected] May 4, 2012 Abstract Full intuitionistic Zermelo-Fraenkel set theory, IZF, is obtained from constructive Zermelo- Fraenkel set theory, CZF, by adding the full separation axiom scheme and the power set axiom.
    [Show full text]
  • SET THEORY for CATEGORY THEORY 3 the Category Is Well-Powered, Meaning That Each Object Has Only a Set of Iso- Morphism Classes of Subobjects
    SET THEORY FOR CATEGORY THEORY MICHAEL A. SHULMAN Abstract. Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible. In this expository paper we summarize and compare a num- ber of such “set-theoretic foundations for category theory,” and describe their implications for the everyday use of category theory. We assume the reader has some basic knowledge of category theory, but little or no prior experience with formal logic or set theory. 1. Introduction Since its earliest days, category theory has had to deal with set-theoretic ques- tions. This is because unlike in most fields of mathematics outside of set theory, questions of size play an essential role in category theory. A good example is Freyd’s Special Adjoint Functor Theorem: a functor from a complete, locally small, and well-powered category with a cogenerating set to a locally small category has a left adjoint if and only if it preserves small limits. This is always one of the first results I quote when people ask me “are there any real theorems in category theory?” So it is all the more striking that it involves in an essential way notions like ‘locally small’, ‘small limits’, and ‘cogenerating set’ which refer explicitly to the difference between sets and proper classes (or between small sets and large sets). Despite this, in my experience there is a certain amount of confusion among users and students of category theory about its foundations, and in particular about what constructions on large categories are or are not possible.
    [Show full text]
  • An Introduction to Set Theory
    AN INTRODUCTION TO SET THEORY Professor William A. R. Weiss November 21, 2014 2 Contents 0 Introduction 7 1 LOST 11 2 FOUND 23 3 The Axioms of Set Theory 29 4 The Natural Numbers 37 5 The Ordinal Numbers 47 6 Relations and Orderings 59 7 Cardinality 69 8 What's So Real About The Real Numbers? 79 9 Ultrafilters Are Useful 87 3 4 CONTENTS 10 The Universe 97 11 Reflection 103 12 Elementary Submodels 123 13 Constructibility 139 14 Appendices 155 .1 The Axioms of ZFC . 155 .2 Tentative Axioms . 156 CONTENTS 5 Preface These notes for a graduate course in set theory are on their way to be- coming a book. They originated as handwritten notes in a course at the University of Toronto given by Prof. William Weiss. Cynthia Church pro- duced the first electronic copy in December 2002. James Talmage Adams produced a major revision in February 2005. The manuscript has seen many changes since then, often due to generous comments by students, each of whom I here thank. Chapters 1 to 11 are now close to final form. Chapters 12 and 13 are quite readable, but should not be considered as a final draft. One more chapter will be added. 6 CONTENTS Chapter 0 Introduction Set Theory is the true study of infinity. This alone assures the subject of a place prominent in human culture. But even more, Set Theory is the milieu in which mathematics takes place today. As such, it is expected to provide a firm foundation for all the rest of mathematics.
    [Show full text]