Extending the Language of Set Theory
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Infinitary Logic and Inductive Definability Over Finite Structures
University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science November 1991 Infinitary Logic and Inductive Definability Over Finite Structures Anuj Dawar University of Pennsylvania Steven Lindell University of Pennsylvania Scott Weinstein University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/cis_reports Recommended Citation Anuj Dawar, Steven Lindell, and Scott Weinstein, "Infinitary Logic and Inductive Definability Over Finite Structures", . November 1991. University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-91-97. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/365 For more information, please contact [email protected]. Infinitary Logic and Inductive Definability Over Finite Structures Abstract The extensions of first-order logic with a least fixed point operators (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [AV91b] investigated the relation of these two logics in the absence of an ordering, using a mchine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each ω formula has a bounded number of variables, L ∞ω (see, for instance, [KV90]). We present a treatment of the results in [AV91b] from this point of view. In particular, we show that we can write a formula of FO + LFP and P from ordered structures to classes of structures where every element is definable. -
⊥N AS an ABSTRACT ELEMENTARY CLASS We Show
⊥N AS AN ABSTRACT ELEMENTARY CLASS JOHN T. BALDWIN, PAUL C. EKLOF, AND JAN TRLIFAJ We show that the concept of an Abstract Elementary Class (AEC) provides a unifying notion for several properties of classes of modules and discuss the stability class of these AEC. An abstract elementary class consists of a class of models K and a strengthening of the notion of submodel ≺K such that (K, ≺K) satisfies ⊥ the properties described below. Here we deal with various classes ( N, ≺N ); the precise definition of the class of modules ⊥N is given below. A key innovation is ⊥ that A≺N B means A ⊆ B and B/A ∈ N. We define in the text the main notions used here; important background defini- tions and proofs from the theory of modules can be found in [EM02] and [GT06]; concepts of AEC are due to Shelah (e.g. [She87]) but collected in [Bal]. The surprising fact is that some of the basic model theoretic properties of the ⊥ ⊥ class ( N, ≺N ) translate directly to algebraic properties of the class N and the module N over the ring R that have previously been studied in quite a different context (approximation theory of modules, infinite dimensional tilting theory etc.). Our main results, stated with increasingly strong conditions on the ring R, are: ⊥ Theorem 0.1. (1) Let R be any ring and N an R–module. If ( N, ≺N ) is an AEC then N is a cotorsion module. Conversely, if N is pure–injective, ⊥ then the class ( N, ≺N ) is an AEC. (2) Let R be a right noetherian ring and N be an R–module of injective di- ⊥ ⊥ mension ≤ 1. -
Arxiv:2010.02145V2 [Math.LO] 25 Feb 2021 Infinitary Logics and Abstract
Infinitary Logics and Abstract Elementary Classes Saharon Shelah1 and Andrés Villaveces2 1Hebrew University of Jerusalem / Rutgers University 2Universidad Nacional de Colombia - Bogotá March 1, 2021 Abstract We prove that every abstract elementary class (a.e.c.) with LST number κ and vocabulary τ of cardinality 6 κ can be axiomatized in L +++ + the logic i2(κ) ,κ (τ). In this logic an a.e.c. is therefore an EC class rather than merely a PC class. This constitutes a major improvement on the level of definability previously given by the Presentation Theorem. As part of our proof, we define the canonical tree S = SK of an a.e.c. K. This turns out to be an interesting combinatorial object of the class, beyond the aim of our theorem. Furthermore, we study a connection between the sentences defining an a.e.c. and the relatively new infinitary 1 logic Lλ. arXiv:2010.02145v2 [math.LO] 25 Feb 2021 Introduction Given an abstract elementary class (a.e.c.) K, in vocabulary τ of size 6 κ = LST(K), we do two main things: • We provide an infinitary sentence in the same vocabulary τ of the a.e.c. that axiomatizes K. Research of both authors partially supported by NSF grant no: DMS 1833363. Research of first author partially supported by Israel Science Foundation (ISF) grant no: 1838/19. 1 Infinitary Logics and Abstract Elementary Classes 2 • We also provide a version of the “Tarski-Vaught-criterion,” adapted to a.e.c.’s: when M1 ⊆ M2, for M1, M2 ∈ K, we will provide nec- essary and sufficient syntactic conditions for M1 ≺K M2. -
Calibrating Determinacy Strength in Borel Hierarchies
University of California Los Angeles Calibrating Determinacy Strength in Borel Hierarchies A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Sherwood Julius Hachtman 2015 c Copyright by Sherwood Julius Hachtman 2015 Abstract of the Dissertation Calibrating Determinacy Strength in Borel Hierarchies by Sherwood Julius Hachtman Doctor of Philosophy in Mathematics University of California, Los Angeles, 2015 Professor Itay Neeman, Chair We study the strength of determinacy hypotheses in levels of two hierarchies of subsets of 0 0 1 Baire space: the standard Borel hierarchy hΣαiα<ω1 , and the hierarchy of sets hΣα(Π1)iα<ω1 in the Borel σ-algebra generated by coanalytic sets. 0 We begin with Σ3, the lowest level at which the strength of determinacy had not yet been characterized in terms of a natural theory. Building on work of Philip Welch [Wel11], 0 [Wel12], we show that Σ3 determinacy is equivalent to the existence of a β-model of the 1 axiom of Π2 monotone induction. 0 For the levels Σ4 and above, we prove best-possible refinements of old bounds due to Harvey Friedman [Fri71] and Donald A. Martin [Mar85, Mar] on the strength of determinacy in terms of iterations of the Power Set axiom. We introduce a novel family of reflection 0 principles, Π1-RAPα, and prove a level-by-level equivalence between determinacy for Σ1+α+3 and existence of a wellfounded model of Π1-RAPα. For α = 0, we have the following concise 0 result: Σ4 determinacy is equivalent to the existence of an ordinal θ so that Lθ satisfies \P(!) exists, and all wellfounded trees are ranked." 0 We connect our result on Σ4 determinacy to work of Noah Schweber [Sch13] on higher order reverse mathematics. -
Arxiv:1901.02074V1 [Math.LO] 4 Jan 2019 a Xoskona Lrecrias Ih Eal Ogv Entv a Definitive a Give T to of Able Basis Be the Might Cardinals” [17]
GENERIC LARGE CARDINALS AS AXIOMS MONROE ESKEW Abstract. We argue against Foreman’s proposal to settle the continuum hy- pothesis and other classical independent questions via the adoption of generic large cardinal axioms. Shortly after proving that the set of all real numbers has a strictly larger car- dinality than the set of integers, Cantor conjectured his Continuum Hypothesis (CH): that there is no set of a size strictly in between that of the integers and the real numbers [1]. A resolution of CH was the first problem on Hilbert’s famous list presented in 1900 [19]. G¨odel made a major advance by constructing a model of the Zermelo-Frankel (ZF) axioms for set theory in which the Axiom of Choice and CH both hold, starting from a model of ZF. This showed that the axiom system ZF, if consistent on its own, could not disprove Choice, and that ZF with Choice (ZFC), a system which suffices to formalize the methods of ordinary mathematics, could not disprove CH [16]. It remained unknown at the time whether models of ZFC could be found in which CH was false, but G¨odel began to suspect that this was possible, and hence that CH could not be settled on the basis of the normal methods of mathematics. G¨odel remained hopeful, however, that new mathemati- cal axioms known as “large cardinals” might be able to give a definitive answer on CH [17]. The independence of CH from ZFC was finally solved by Cohen’s invention of the method of forcing [2]. Cohen’s method showed that ZFC could not prove CH either, and in fact could not put any kind of bound on the possible number of cardinals between the sizes of the integers and the reals. -
Structural Logic and Abstract Elementary Classes with Intersections
STRUCTURAL LOGIC AND ABSTRACT ELEMENTARY CLASSES WITH INTERSECTIONS WILL BONEY AND SEBASTIEN VASEY Abstract. We give a syntactic characterization of abstract elementary classes (AECs) closed under intersections using a new logic with a quantifier for iso- morphism types that we call structural logic: we prove that AECs with in- tersections correspond to classes of models of a universal theory in structural logic. This generalizes Tarski's syntactic characterization of universal classes. As a corollary, we obtain that any AEC closed under intersections with count- able L¨owenheim-Skolem number is axiomatizable in L1;!(Q), where Q is the quantifier \there exists uncountably many". Contents 1. Introduction 1 2. Structural quantifiers 3 3. Axiomatizing abstract elementary classes with intersections 7 4. Equivalence vs. Axiomatization 11 References 13 1. Introduction 1.1. Background and motivation. Shelah's abstract elementary classes (AECs) [She87,Bal09,She09a,She09b] are a semantic framework to study the model theory of classes that are not necessarily axiomatized by an L!;!-theory. Roughly speak- ing (see Definition 2.4), an AEC is a pair (K; ≤K) satisfying some of the category- theoretic properties of (Mod(T ); ), for T an L!;!-theory. This encompasses classes of models of an L1;! sentence (i.e. infinite conjunctions and disjunctions are al- lowed), and even L1;!(hQλi ii<α) theories, where Qλi is the quantifier \there exists λi-many". Since the axioms of AECs do not contain any axiomatizability requirement, it is not clear whether there is a natural logic whose class of models are exactly AECs. More precisely, one can ask whether there is a natural abstract logic (in the sense of Barwise, see the survey [BFB85]) so that classes of models of theories in that logic are AECs and any AEC is axiomatized by a theory in that logic. -
Peter Koellner
Peter Koellner Department of Philosophy 50 Follen St. 320 Emerson Hall, Harvard University Cambridge, MA, 02138 Cambridge, MA 02138 (617) 821-4688 (617) 495-3970 [email protected] Employment Professor of Philosophy, Harvard University, (2010{) John L. Loeb Associate Professor of the Humanities, Harvard University, (2008{2010) Assistant Professor of Philosophy, Harvard University, (2003{2008) Education Ph.D. in Philosophy Massachusetts Institute of Technology, 2003 (Minor: Mathematical Logic) Research in Logic University of California, Berkeley Logic Group: 1998{2002 M.A. in Philosophy University of Western Ontario, 1996 B.A. in Philosophy University of Toronto, 1995 Research Interests Primary: Set Theory, Foundations of Mathematics, Philosophy of Mathematics Secondary: Foundations of Physics, Early Analytic Philosophy, History of Philosophy and the Exact Sciences Academic Awards and Honors • John Templeton Foundation Research Grant: Exploring the Frontiers of Incompleteness; 2011-2013 |$195,029 • Kurt G¨odelCentenary Research Fellowship Prize; 2008{2011 |$130,000 • John Templeton Foundation Research Grant: Exploring the Infinite (with W. Hugh Woodin); 2008{2009 |$50,000 • Clarke-Cooke Fellowships; 2005{2006, 2006{2007 • Social Sciences and Humanities Research Council of Canada Scholarship, 1996{2000 (tenure 1998{2000: Berkeley) Peter Koellner Curriculum Vitae 2 of 4 Teaching Experience Courses taught at Harvard: Set Theory (Phil 143, Fall 2003) Set Theory: Large Cardinals from Determinacy (Math 242, Fall 2004) Set Theory: Inner Model -
The Higher Infinite in Proof Theory
The Higher Infinite in Proof Theory∗y Michael Rathjen Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom 1 Introduction The higher infinite usually refers to the lofty reaches of Cantor's paradise, notably to the realm of large cardinals whose existence cannot be proved in the established for- malisation of Cantorian set theory, i.e. Zermelo-Fraenkel set theory with the axiom of choice. Proof theory, on the other hand, is commonly associated with the manipula- tion of syntactic objects, that is finite objects par excellence. However, finitary proof theory became already infinitary in the 1950's when Sch¨uttere-obtained Gentzen's ordinal analysis for number theory in a particular transparent way through the use of an infinitary proof system with the so-called !-rule (cf. [54]). Nowadays one even finds vestiges of large cardinals in ordinal-theoretic proof theory. Large cardinals have worked their way down through generalized recursion (in the shape of recursively large ordinals) to proof theory wherein they appear in the definition procedures of so-called collapsing functions which give rise to ordinal representation systems. The surprising use of ordinal representation systems employing \names" for large cardinals in current proof-theoretic ordinal analyses is the main theme of this paper. The exposition here diverges from the presentation given at the conference in two regards. Firstly, the talk began with a broad introduction, explaining the current rationale and goals of ordinal-theoretic proof theory, which take the place of the original Hilbert Program. Since this part of the talk is now incorporated in the first two sections of the BSL-paper [48] there is no point in reproducing it here. -
First Order Logic, Fixed Point Logic and Linear Order Anuj Dawar University of Pennsylvania
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ScholarlyCommons@Penn University of Pennsylvania ScholarlyCommons IRCS Technical Reports Series Institute for Research in Cognitive Science November 1995 First Order Logic, Fixed Point Logic and Linear Order Anuj Dawar University of Pennsylvania Steven Lindell University of Pennsylvania Scott einsW tein University of Pennsylvania, [email protected] Follow this and additional works at: http://repository.upenn.edu/ircs_reports Dawar, Anuj; Lindell, Steven; and Weinstein, Scott, "First Order Logic, Fixed Point Logic and Linear Order" (1995). IRCS Technical Reports Series. 145. http://repository.upenn.edu/ircs_reports/145 University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-95-30. This paper is posted at ScholarlyCommons. http://repository.upenn.edu/ircs_reports/145 For more information, please contact [email protected]. First Order Logic, Fixed Point Logic and Linear Order Abstract The Ordered conjecture of Kolaitis and Vardi asks whether fixed-point logic differs from first-order logic on every infinite class of finite ordered structures. In this paper, we develop the tool of bounded variable element types, and illustrate its application to this and the original conjectures of McColm, which arose from the study of inductive definability and infinitary logic on proficient classes of finite structures (those admitting an unbounded induction). In particular, for a class of finite structures, we introduce a compactness notion which yields a new proof of a ramified version of McColm's second conjecture. Furthermore, we show a connection between a model-theoretic preservation property and the Ordered Conjecture, allowing us to prove it for classes of strings (colored orderings). -
Large Cardinals in Set Theory from Their Beginnings
208 BOOK REVIEWS MASON, L. J. and WOODHOUSE, N. M. J. Integrability, self-duality and twistor theory (London Mathematical Society Monographs (New Series) No. 15, Clarendon Press, Oxford, 1996), x + 364pp., 0 19 853498 1, (hardback) £45. Integrable systems of PDEs have come a long way for a subject where the object of study lacks a definition. In medicine it is common to define a condition by saying that to qualify a patient must exhibit (say) four out of six given symptoms and perhaps we should do something analogous to define integrability of a system of PDEs, the symptoms being things like: the existence of nonlinear "superpositions" of solutions; existence of "sufficient" constants of the motion; existence of a "Lax pair"; satisfying analogues of the Painleve condition for ODEs; etc. The unifying idea behind this book is the suggestion that one particular feature that is particularly important in integrability is the existence of a "twistor construction" for the system. The authors' hope (I am perhaps putting words into their mouths here) is that, if one knew how to say this correctly, it might serve as a single feature characterising integrable systems. The chain of ideas which has lead to this is as follows. One example of an integrable system is the self-dual Yang-Mills equations for a connection on a vector bundle over R4. One has a twistor construction for this system (the Ward correspondence) which puts solutions of this system in one-to-one correspondence with a certain set of holomorphic vector bundles over a subset of complex projective 3-space. -
Paul B. Larson a BRIEF HISTORY of DETERMINACY §1. Introduction
Paul B. Larson A BRIEF HISTORY OF DETERMINACY x1. Introduction. Determinacy axioms are statements to the effect that certain games are determined, in that each player in the game has an optimal strategy. The commonly accepted axioms for mathematics, the Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC; see [??, ??]), im- ply the determinacy of many games that people actually play. This applies in particular to many games of perfect information, games in which the players alternate moves which are known to both players, and the out- come of the game depends only on this list of moves, and not on chance or other external factors. Games of perfect information which must end in finitely many moves are determined. This follows from the work of Ernst Zermelo [??], D´enesK}onig[??] and L´aszl´oK´almar[??], and also from the independent work of John von Neumann and Oskar Morgenstern (in their 1944 book, reprinted as [??]). As pointed out by Stanis law Ulam [??], determinacy for games of perfect information of a fixed finite length is essentially a theorem of logic. If we let x1,y1,x2,y2,::: ,xn,yn be variables standing for the moves made by players player I (who plays x1,::: ,xn) and player II (who plays y1,::: ,yn), and A (consisting of sequences of length 2n) is the set of runs of the game for which player I wins, the statement (1) 9x18y1 ::: 9xn8ynhx1; y1; : : : ; xn; yni 2 A essentially asserts that the first player has a winning strategy in the game, and its negation, (2) 8x19y1 ::: 8xn9ynhx1; y1; : : : ; xn; yni 62 A essentially asserts that the second player has a winning strategy.1 We let ! denote the set of natural numbers 0; 1; 2;::: ; for brevity we will often refer to the members of this set as \integers". -
Declarative/Logic-Based Computational Cognitive Modeling∗
Declarative/Logic-Based Computational Cognitive Modeling∗ Selmer Bringsjord Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department of Computer Science Rensselaer Polytechnic Institute (RPI) Troy NY 12180 USA [email protected] draft of 031607.0108NY Abstract This chapter is an esemplastic systematization of declarative computational cognitive modeling, a field that cuts across cognitive modeling based on cognitive architectures (such as ACT-R, Soar, and Clar- ion), human-level artificial intelligence (AI), logic itself, and psychology of reasoning (especially of the computational kind). The hallmarks of declarative computational cognitive modeling are the following two intertwined constraints: (1) The central units of information used in the approach are (at least in significant part) declarative in nature, and the central process carried out over these units is inference. (2) The approach to modeling the mind is top-down, rather than bottom-up. (These two points are inter- connected because once one commits to (1), (2) becomes quite unavoidable, since bottom-up processing in the brain, as reflected in relevant formalisms (e.g., artificial neural networks), is based on units of infor- mation that are numerical, not declarative.) The systematization of declarative computational cognitive modeling is achieved by using formal logic, and hence declarative computational cognitive modeling, from the formal perspective, becomes logic-based computational cognitive modeling (LCCM). The chapter covers some prior research that falls under LCCM; this research has been carried out by such thinkers as Johnson-Laird, Langley, Rips, Simon, and Sun. The material that follows is introductory in nature, and self-contained; it assumes only a modicum of previous exposure to discrete mathematics and computability theory.