DOCSLIB.ORG

Measurable Cardinal Wikipedia Contents

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Measurable Cardinal Wikipedia Contents Measurable cardinal Wikipedia Contents 1 Atom (measure theory) 1 1.1 Definition ............................................... 1 1.2 Examples ............................................... 1 1.3 Non-atomic measures ......................................... 1 1.4 See also ................................................ 2 1.5 Notes ................................................. 2 1.6 References ............................................... 2 2 Axiom of determinacy 3 2.1 Types of game that are determined .................................. 3 2.2 Incompatibility of the axiom of determinacy with the axiom of choice ................ 3 2.3 Infinite logic and the axiom of determinacy .............................. 4 2.4 Large cardinals and the axiom of determinacy ............................ 4 2.5 See also ................................................ 4 2.6 References ............................................... 5 2.7 Further reading ............................................ 5 3 Cardinal number 6 3.1 History ................................................. 6 3.2 Motivation .............................................. 8 3.3 Formal definition ........................................... 9 3.4 Cardinal arithmetic .......................................... 10 3.4.1 Successor cardinal ...................................... 10 3.4.2 Cardinal addition ....................................... 10 3.4.3 Cardinal multiplication .................................... 11 3.4.4 Cardinal exponentiation ................................... 11 3.5 The continuum hypothesis ....................................... 12 3.6 See also ................................................ 12 3.7 References ............................................... 12 3.8 External links ............................................. 13 4 Class (set theory) 14 4.1 Examples ............................................... 14 i ii CONTENTS 4.2 Paradoxes ............................................... 14 4.3 Classes in formal set theories ..................................... 14 4.4 References ............................................... 15 4.5 External links ............................................. 15 5 Club filter 16 5.1 References ............................................... 16 6 Club set 17 6.1 Formal definition ........................................... 17 6.2 The closed unbounded filter ...................................... 17 6.3 See also ................................................ 18 6.4 References ............................................... 18 7 Continuum hypothesis 19 7.1 Cardinality of infinite sets ....................................... 19 7.2 Independence from ZFC ........................................ 20 7.3 Arguments for and against CH .................................... 20 7.4 The generalized continuum hypothesis ................................ 21 7.4.1 Implications of GCH for cardinal exponentiation ....................... 22 7.5 See also ................................................ 22 7.6 References ............................................... 22 7.7 External links ............................................. 23 8 Critical point (set theory) 24 8.1 References ............................................... 24 9 Dana Scott 25 9.1 Early career .............................................. 25 9.2 University of California, Berkeley, 1960–1963 ............................ 25 9.2.1 Modal and tense logic ..................................... 26 9.3 Stanford, Amsterdam and Princeton, 1963–1972 ........................... 26 9.4 Oxford University, 1972–1981 .................................... 26 9.4.1 Semantics of programming languages ............................. 26 9.5 Carnegie Mellon University 1981–2003 ................................ 27 9.6 See also ................................................ 27 9.7 Bibliography .............................................. 27 9.7.1 Works by Scott ........................................ 27 9.7.2 Other works .......................................... 27 9.8 References ............................................... 27 9.9 External links ............................................. 28 10 Diagonal intersection 29 10.1 See also ................................................ 29 CONTENTS iii 10.2 References ............................................... 29 11 Elementary equivalence 30 11.1 Elementarily equivalent structures ................................... 30 11.2 Elementary substructures and elementary extensions ......................... 30 11.3 Tarski–Vaught test ........................................... 31 11.4 Elementary embeddings ........................................ 31 11.5 References .............................................. 31 12 Equiconsistency 32 12.1 Consistency .............................................. 32 12.2 Consistency strength ......................................... 32 12.3 See also ................................................ 33 12.4 References ............................................... 33 13 Extender (set theory) 34 13.1 Formal definition of an extender .................................... 34 13.2 Defining an extender from an elementary embedding ......................... 34 13.3 References ............................................... 35 14 Fodor’s lemma 36 14.1 Proof ................................................. 36 14.2 Fodor’s lemma for trees ........................................ 36 14.3 References .............................................. 36 15 Huge cardinal 37 15.1 Variants ................................................ 37 15.2 Consistency strength ......................................... 38 15.3 ω-huge cardinals ............................................ 38 15.4 See also ................................................ 38 15.5 References .............................................. 38 16 Inaccessible cardinal 39 16.1 Models and consistency ........................................ 39 16.2 Existence of a proper class of inaccessibles .............................. 40 16.3 α-inaccessible cardinals and hyper-inaccessible cardinals ....................... 40 16.4 Two model-theoretic characterisations of inaccessibility ....................... 40 16.5 See also ................................................ 41 16.6 References .............................................. 41 17 Ineffable cardinal 42 17.1 References ............................................... 42 18 Intersection (set theory) 43 iv CONTENTS 18.1 Basic definition ............................................ 43 18.1.1 Intersecting and disjoint sets ................................. 45 18.2 Arbitrary intersections ......................................... 46 18.3 Nullary intersection .......................................... 47 18.4 See also ................................................ 48 18.5 References ............................................... 48 18.6 Further reading ............................................ 48 18.7 External links ............................................. 48 19 Lebesgue measure 49 19.1 Definition ............................................... 49 19.1.1 Intuition ........................................... 49 19.2 Examples ............................................... 50 19.3 Properties ............................................... 50 19.4 Null sets ................................................ 51 19.5 Construction of the Lebesgue measure ................................ 52 19.6 Relation to other measures ...................................... 52 19.7 See also ................................................ 53 19.8 References .............................................. 53 20 Measurable cardinal 54 20.1 Measurable .............................................. 54 20.2 Real-valued measurable ........................................ 55 20.3 See also ................................................ 55 20.4 References .............................................. 55 21 Measure (mathematics) 56 21.1 Definition ............................................... 56 21.2 Examples ............................................... 57 21.3 Properties ............................................... 57 21.3.1 Monotonicity ......................................... 58 21.3.2 Measures of infinite unions of measurable sets ........................ 58 21.3.3 Measures of infinite intersections of measurable sets ..................... 58 21.4 Sigma-finite measures ......................................... 58 21.5 Completeness ............................................. 59 21.6 Additivity ............................................... 59 21.7 Non-measurable sets .......................................... 59 21.8 Generalizations ............................................ 59 21.9 See also ................................................ 60 21.10References ............................................... 60 21.11Bibliography ............................................. 61 21.12External links ............................................. 61 CONTENTS v 22 Mitchell order 64 22.1 References ............................................... 64 23 Model theory 65 23.1 Branches of model theory ....................................... 65 23.2 Universal algebra ........................................... 66 23.3 Finite model theory .......................................... 67 23.4 First-order logic ............................................ 67 23.5 Axiomatizability, elimination of quantifiers, and model-completeness ................ 68
Load more

Recommended publications
  • Diagonal Cauchy Spaces D.C
    BULL. AUSTRAL. MATH. SOC. 54E15, 54D10, 54D35, 54A20 VOL. 54 (1996) [255-265] DIAGONAL CAUCHY SPACES D.C. KENT AND G.D. RICHARDSON A diagonal condition is defined which internally characterises those Cauchy spaces which have topological completions. The T? diagonal Cauchy spaces allow both a finest and a coarsest T2 diagonal completion. The former is a completion functor, while the latter preserves uniformisability and has an extension property relative to ^-continuous maps. INTRODUCTION In 1954, Kowalsky [2] defined a diagonal axiom for convergence spaces subject to which every pretopological space is topological. In 1967, Cook and Fischer [1] gave a stronger version of the Kowalsky axiom relative to which a larger class of convergence spaces was shown to be topological. In [4], we showed that any convergence space satisfying the Cook-Fischer diagonal axiom is topological. In this paper, we introduce a diagonal axiom for Cauchy spaces which reduces to the Cook-Fischer axiom when the Cauchy space is complete. A diagonal Cauchy space is one which satisfies this axiom. We show that a Cauchy space is diagonal if and only if every Cauchy equivalence class contains a smallest Cauchy filter, and this smallest Cauchy filter has a base of open sets. Equivalently, a Cauchy space is diagonal if and only if it allows a diagonal (that is, topological) completion. The category of diagoaal Cauchy spaces and Cauchy continuous maps is shown to be a topological category. It is shown that a Ti diagonal Cauchy space has both a finest and a coarsest T2 diagonal completion. The "fine diagonal completion" determines a completion functor on the category of T2, diagonal Cauchy spaces.
    [Show full text]
  • $ G $-Metrizable Spaces and the Images of Semi-Metric Spaces
    Czechoslovak Mathematical Journal Ying Ge; Shou Lin g-metrizable spaces and the images of semi-metric spaces Czechoslovak Mathematical Journal, Vol. 57 (2007), No. 4, 1141–1149 Persistent URL: http://dml.cz/dmlcz/128231 Terms of use: © Institute of Mathematics AS CR, 2007 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Czechoslovak Mathematical Journal, 57 (132) (2007), 1141–1149 g-METRIZABLE SPACES AND THE IMAGES OF SEMI-METRIC SPACES Ying Ge, Jiangsu, Shou Lin, Fujian (Received November 8, 2005) Abstract. In this paper, we prove that a space X is a g-metrizable space if and only if X is a weak-open, π and σ-image of a semi-metric space, if and only if X is a strong sequence-covering, quotient, π and mssc-image of a semi-metric space, where “semi-metric” can not be replaced by “metric”. Keywords: g-metrizable spaces, sn-metrizable spaces, weak-open mappings, strong sequence-covering mappings, quotient mappings, π-mappings, σ-mappings, mssc-mappings MSC 2000 : 54C10, 54D55, 54E25, 54E35, 54E40 1. Introduction g-metrizable spaces as a generalization of metric spaces have many important properties [17]. To characterize g-metrizable spaces as certain images of metric spaces is an interesting question in the theory of generalized metric spaces, and many “nice” characterizations of g-metrizable spaces have been obtained ([6], [8], [7], [13], [18], [19]).
    [Show full text]
  • On the Universality of Memcomputing Machines Yan Ru Pei, Fabio L
    1 On the Universality of Memcomputing Machines Yan Ru Pei, Fabio L. Traversa, and Massimiliano Di Ventra Abstract—Universal memcomputing machines (UMMs) [IEEE is related to the topology (or architecture) of the network of Trans. Neural Netw. Learn. Syst. 26, 2702 (2015)] represent memory units (memprocessors). It means the machine has a novel computational model in which memory (time non- access, at any given time, to more information (precisely locality) accomplishes both tasks of storing and processing of information. UMMs have been shown to be Turing-complete, originating from the topology of the network) than what is namely they can simulate any Turing machine. In this paper, we available if the memprocessors were not connected to each first introduce a novel set-theory approach to compare different other. Of course, this information overhead is not necessarily computational models, and use it to recover the previous results stored by the machine (the stored information is the Shannon on Turing completeness of UMMs. We then relate UMMs directly one) [4]. Nevertheless, with appropriate topologies, hence with to liquid-state machines (or “reservoir computing”) and quantum machines (“quantum computing”). We show that UMMs can appropriate information overhead, UMMs can solve complex simulate both types of machines, hence they are both “liquid-” or problems very efficiently [4]. Finally, functional polymor- “reservoir-complete” and “quantum-complete”. Of course, these phism means that the machine is able to compute different statements pertain only to the type of problems these machines functions by simply applying the appropriate input signals [4], can solve, and not to the amount of resources required for such without modifying the topology of the machine network.
    [Show full text]
  • Two Moore Manifolds
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 51 (1993) 27-39 27 North-Holland Two Moore manifolds I.J. Tree* Department ofMathematics, University of North Texas, Denton, TX 76203, USA W.S. Watson Department of Mathematics, University of York, North York, Ont., Canada M3J IP3 Received 23 August 1991 Revised 27 April 1992 Abstract Tree, I.J. and W.S. Watson, Two Moore manifolds, Topology and its Applications 51 (1993) 27-39. Reed and Zenor have shown that locally connected, locally compact, normal Moore spaces are metrizable. The first of the two examples presented is a locally connected, locally compact, pseudonormal nonmetrizable Moore space. The second is a locally connected, locally compact, pseudocompact nonmetrizable Moore space and can be constructed assuming the Continuum Hypothesis. Therefore normality in the Reed-Zenor theorem cannot be replaced by pseudonor- mality or (consistently) pseudocompactness. Both spaces can be modified in such a way that they are manifolds. Keywords: Pseudocompact, pseudonormal, A-set, Cook set. AMS (MOS) Subj. Class.: Primary 54020; secondary 54830, 57N05. 1. Introduction We know that, on one hand, the truth of the normal Moore space conjecture-the statement that every normal Moore space is metrizable-is intimately tied to the existence of certain large cardinals [6]. On the other hand, in particular extended versions of set theory there exist normal nonmetrizable Moore spaces. Assuming V= L, every locally compact normal Moore space is metrizable [5]; but under MA+CH, there exist locally compact normal nonmetrizable Moore spaces [18].
    [Show full text]
  • Topology Proceedings
    Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT °c by Topology Proceedings. All rights reserved. TOPOLOGY PROCEEDINGS Volume 30, No.2, 2006 Pages 523-532 THE ROLE OF NORMALITY IN THE METRIZATION OF MOORE SPACES H.H. HUNG Abstract. We take away the property of normality from Bing’s collectionwise normality in his celebrated theorem on the metrization of Moore Spaces, and we weaken the no- tion of normality itself in the equally celebrated theorem of F.B. Jones for the separable case under the assumption of Lusin’s. Normality conjectured to be sufficient, if not indis- pensible, for the metrization of Moore spaces, its role in the theory is hereby questioned. Traylor’s theorem that sepa- rable metacompact Moore spaces are metrizable is also im- proved on. There are also two generalizations of Bing’s gen- eral metrization theorem. It has been conjectured that if a Moore space fails to be metriz- able, it cannot be normal, an opinion no doubt encouraged by the near-conclusive result for the separable case: Normal separa- ble Moore spaces are metrizable, if 2ω < 2ω1 (Theorem 5 of [10]). Strengthening the property of normality to that of collectionwise normality in the conjecture, Bing [1] was able to give an affirmative result, arguably (see e.g. §3 of [13]) the most important theorem on the subject. Taking the property of normality itself out of that of collectionwise normality, I was able in [5] to generalize Bing.
    [Show full text]
  • The Open Handbook of Formal Epistemology
    THEOPENHANDBOOKOFFORMALEPISTEMOLOGY Richard Pettigrew &Jonathan Weisberg,Eds. THEOPENHANDBOOKOFFORMAL EPISTEMOLOGY Richard Pettigrew &Jonathan Weisberg,Eds. Published open access by PhilPapers, 2019 All entries copyright © their respective authors and licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. LISTOFCONTRIBUTORS R. A. Briggs Stanford University Michael Caie University of Toronto Kenny Easwaran Texas A&M University Konstantin Genin University of Toronto Franz Huber University of Toronto Jason Konek University of Bristol Hanti Lin University of California, Davis Anna Mahtani London School of Economics Johanna Thoma London School of Economics Michael G. Titelbaum University of Wisconsin, Madison Sylvia Wenmackers Katholieke Universiteit Leuven iii For our teachers Overall, and ultimately, mathematical methods are necessary for philosophical progress. — Hannes Leitgeb There is no mathematical substitute for philosophy. — Saul Kripke PREFACE In formal epistemology, we use mathematical methods to explore the questions of epistemology and rational choice. What can we know? What should we believe and how strongly? How should we act based on our beliefs and values? We begin by modelling phenomena like knowledge, belief, and desire using mathematical machinery, just as a biologist might model the fluc- tuations of a pair of competing populations, or a physicist might model the turbulence of a fluid passing through a small aperture. Then, we ex- plore, discover, and justify the laws governing those phenomena, using the precision that mathematical machinery affords. For example, we might represent a person by the strengths of their beliefs, and we might measure these using real numbers, which we call credences. Having done this, we might ask what the norms are that govern that person when we represent them in that way.
    [Show full text]
  • Axiomatic Set Teory P.D.Welch
    Axiomatic Set Teory P.D.Welch. August 16, 2020 Contents Page 1 Axioms and Formal Systems 1 1.1 Introduction 1 1.2 Preliminaries: axioms and formal systems. 3 1.2.1 The formal language of ZF set theory; terms 4 1.2.2 The Zermelo-Fraenkel Axioms 7 1.3 Transfinite Recursion 9 1.4 Relativisation of terms and formulae 11 2 Initial segments of the Universe 17 2.1 Singular ordinals: cofinality 17 2.1.1 Cofinality 17 2.1.2 Normal Functions and closed and unbounded classes 19 2.1.3 Stationary Sets 22 2.2 Some further cardinal arithmetic 24 2.3 Transitive Models 25 2.4 The H sets 27 2.4.1 H - the hereditarily finite sets 28 2.4.2 H - the hereditarily countable sets 29 2.5 The Montague-Levy Reflection theorem 30 2.5.1 Absoluteness 30 2.5.2 Reflection Theorems 32 2.6 Inaccessible Cardinals 34 2.6.1 Inaccessible cardinals 35 2.6.2 A menagerie of other large cardinals 36 3 Formalising semantics within ZF 39 3.1 Definite terms and formulae 39 3.1.1 The non-finite axiomatisability of ZF 44 3.2 Formalising syntax 45 3.3 Formalising the satisfaction relation 46 3.4 Formalising definability: the function Def. 47 3.5 More on correctness and consistency 48 ii iii 3.5.1 Incompleteness and Consistency Arguments 50 4 The Constructible Hierarchy 53 4.1 The L -hierarchy 53 4.2 The Axiom of Choice in L 56 4.3 The Axiom of Constructibility 57 4.4 The Generalised Continuum Hypothesis in L.
    [Show full text]
  • A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees
    A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees by Ari Meir Brodsky A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto c Copyright 2014 by Ari Meir Brodsky Abstract A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees Ari Meir Brodsky Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2014 Building on early work by Stevo Todorcevic, we develop a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem. Let κ be any infinite regular cardinal, let ξ be any ordinal such that 2jξj < κ, and let k be any natural number. Then <κ 2 non- 2 -special tree ! (κ + ξ)k : This is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above to the cardinal (2<κ)+, the simplest example of a non-(2<κ)-special tree. As a corollary, we obtain a general result for partially ordered sets: Theorem. Let κ be any infinite regular cardinal, let ξ be any ordinal such that 2jξj < κ, and let k be <κ 1 any natural number. Let P be a partially ordered set such that P ! (2 )2<κ .
    [Show full text]
  • Singular Cardinals: from Hausdorff's Gaps to Shelah's Pcf Theory
    SINGULAR CARDINALS: FROM HAUSDORFF’S GAPS TO SHELAH’S PCF THEORY Menachem Kojman 1 PREFACE The mathematical subject of singular cardinals is young and many of the math- ematicians who made important contributions to it are still active. This makes writing a history of singular cardinals a somewhat riskier mission than writing the history of, say, Babylonian arithmetic. Yet exactly the discussions with some of the people who created the 20th century history of singular cardinals made the writing of this article fascinating. I am indebted to Moti Gitik, Ronald Jensen, Istv´an Juh´asz, Menachem Magidor and Saharon Shelah for the time and effort they spent on helping me understand the development of the subject and for many illuminations they provided. A lot of what I thought about the history of singular cardinals had to change as a result of these discussions. Special thanks are due to Istv´an Juh´asz, for his patient reading for me from the Russian text of Alexandrov and Urysohn’s Memoirs, to Salma Kuhlmann, who directed me to the definition of singular cardinals in Hausdorff’s writing, and to Stefan Geschke, who helped me with the German texts I needed to read and sometimes translate. I am also indebted to the Hausdorff project in Bonn, for publishing a beautiful annotated volume of Hausdorff’s monumental Grundz¨uge der Mengenlehre and for Springer Verlag, for rushing to me a free copy of this book; many important details about the early history of the subject were drawn from this volume. The wonderful library and archive of the Institute Mittag-Leffler are a treasure for anyone interested in mathematics at the turn of the 20th century; a particularly pleasant duty for me is to thank the institute for hosting me during my visit in September of 2009, which allowed me to verify various details in the early research literature, as well as providing me the company of many set theorists and model theorists who are interested in the subject.
    [Show full text]
  • Are Large Cardinal Axioms Restrictive?
    Are Large Cardinal Axioms Restrictive? Neil Barton∗ 24 June 2020y Abstract The independence phenomenon in set theory, while perva- sive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper, I argue that whether or not large cardinal axioms count as maximality prin- ciples depends on prior commitments concerning the richness of the subset forming operation. In particular I argue that there is a conception of maximality through absoluteness, on which large cardinal axioms are restrictive. I argue, however, that large cardi- nals are still important axioms of set theory and can play many of their usual foundational roles. Introduction Large cardinal axioms are widely viewed as some of the best candi- dates for new axioms of set theory. They are (apparently) linearly ordered by consistency strength, have substantial mathematical con- sequences for questions independent from ZFC (such as consistency statements and Projective Determinacy1), and appear natural to the ∗Fachbereich Philosophie, University of Konstanz. E-mail: neil.barton@uni- konstanz.de. yI would like to thank David Aspero,´ David Fernandez-Bret´ on,´ Monroe Eskew, Sy-David Friedman, Victoria Gitman, Luca Incurvati, Michael Potter, Chris Scam- bler, Giorgio Venturi, Matteo Viale, Kameryn Williams and audiences in Cambridge, New York, Konstanz, and Sao˜ Paulo for helpful discussion. Two anonymous ref- erees also provided helpful comments, and I am grateful for their input. I am also very grateful for the generous support of the FWF (Austrian Science Fund) through Project P 28420 (The Hyperuniverse Programme) and the VolkswagenStiftung through the project Forcing: Conceptual Change in the Foundations of Mathematics.
    [Show full text]
  • “Strange” Limits
    c Gabriel Nagy “Strange” Limits Notes from the Functional Analysis Course (Fall 07 - Spring 08) Prerequisites: The reader is assumed to be familiar with the Hahn-Banach Theorem and/or the theory of (ultra)filter convergence. References to some of my notes will be added later. Notations. Given a non-empty set S, we denote by `∞(S) the vector space of all bounded R functions x : S → . On occasion an element x ∈ `∞(S) will also be written as an “S-tuple” R R x = (xs)s∈S. If S is infinite, we denote by Pfin(S) the collection of all finite subsets of S. Assuming S is infinite, for an element x = (x ) ∈ `∞(S), we can define the quantities s s∈S R lim sup xs = inf sup xs , F ∈P (S) S fin s∈SrF lim inf xs = sup inf xs . S s∈S F F ∈Pfin(S) r Definition. Suppose S is infinite. A limit operation on S is a linear map Λ : `∞(S) → , R R satisfying the inequalities: lim inf x ≤ Λ(x) ≤ lim sup x , ∀ x = (x ) ∈ `∞(S). (1) s s s s∈S R S S Comment. Of course, the quantities lim supS xs and lim infS xs can be defined for ar- bitrary “S-tuples,” in which case these limits might equal ±∞. In fact, this allows one to use S-tuples which might take infinite values. In other words, one might consider maps x : S → [−∞, ∞]. We will elaborate on his point of view later in this note. With the above terminology, our problem is to construct limit operations.
    [Show full text]
  • Ineffability Within the Limits of Abstraction Alone
    Ineffability within the Limits of Abstraction Alone Stewart Shapiro and Gabriel Uzquiano 1 Abstraction and iteration The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. The gold standard would be a theory as rich and useful as the dominant one, ZFC, Zermelo-Fraenkel set theory with choice, perhaps aug- mented with large cardinal principles. Although the present paper is self-contained, we draw upon recent work in [32] in which we explore the power of a reasonably pure principle of reflection. To establish terminology, the Scottish neo-logicist program is to develop branches of mathematics using abstraction principles in the form: xα = xβ $ α ∼ β where the variables α and β range over items of a certain sort, x is an operator taking items of this sort to objects, and ∼ is an equivalence relation on this sort of item. The standard exemplar, of course, is Hume’s principle: #F = #G ≡ F ≈ G where, as usual, F ≈ G is an abbreviation of the second-order statement that there is a one-one correspondence from F onto G. Hume’s principle, is the main support for what is regarded as a success-story for the Scottish neo-logicist program, at least by its advocates. The search is on to develop more powerful mathematical theories on a similar basis. As witnessed by this volume and a significant portion of the contemporary liter- ature in the philosophy of mathematics, Scottish neo-logicism remains controver- sial, even for arithmetic. There are, for example, issues of Caesar and Bad Com- pany to deal with.
    [Show full text]