Levy and Set Theory
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Set Theory, by Thomas Jech, Academic Press, New York, 1978, Xii + 621 Pp., '$53.00
BOOK REVIEWS 775 BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Number 1, July 1980 © 1980 American Mathematical Society 0002-9904/80/0000-0 319/$01.75 Set theory, by Thomas Jech, Academic Press, New York, 1978, xii + 621 pp., '$53.00. "General set theory is pretty trivial stuff really" (Halmos; see [H, p. vi]). At least, with the hindsight afforded by Cantor, Zermelo, and others, it is pretty trivial to do the following. First, write down a list of axioms about sets and membership, enunciating some "obviously true" set-theoretic principles; the most popular Hst today is called ZFC (the Zermelo-Fraenkel axioms with the axiom of Choice). Next, explain how, from ZFC, one may derive all of conventional mathematics, including the general theory of transfinite cardi nals and ordinals. This "trivial" part of set theory is well covered in standard texts, such as [E] or [H]. Jech's book is an introduction to the "nontrivial" part. Now, nontrivial set theory may be roughly divided into two general areas. The first area, classical set theory, is a direct outgrowth of Cantor's work. Cantor set down the basic properties of cardinal numbers. In particular, he showed that if K is a cardinal number, then 2", or exp(/c), is a cardinal strictly larger than K (if A is a set of size K, 2* is the cardinality of the family of all subsets of A). Now starting with a cardinal K, we may form larger cardinals exp(ic), exp2(ic) = exp(exp(fc)), exp3(ic) = exp(exp2(ic)), and in fact this may be continued through the transfinite to form expa(»c) for every ordinal number a. -
Deduction Systems Based on Resolution, We Limit the Following Considerations to Transition Systems Based on Analytic Calculi
Author’s Address Norbert Eisinger European Computer–Industry Research Centre, ECRC Arabellastr. 17 D-8000 M¨unchen 81 F. R. Germany [email protected] and Hans J¨urgen Ohlbach Max–Planck–Institut f¨ur Informatik Im Stadtwald D-6600 Saarbr¨ucken 11 F. R. Germany [email protected] Publication Notes This report appears as chapter 4 in Dov Gabbay (ed.): ‘Handbook of Logic in Artificial Intelligence and Logic Programming, Volume I: Logical Foundations’. It will be published by Oxford University Press, 1992. Fragments of the material already appeared in chapter two of Bl¨asis & B¨urckert: Deduction Systems in Artificial Intelligence, Ellis Horwood Series in Artificial Intelligence, 1989. A draft version has been published as SEKI Report SR-90-12. The report is also published as an internal technical report of ECRC, Munich. Acknowledgements The writing of the chapters for the handbook has been a highly coordinated effort of all the people involved. We want to express our gratitude for their many helpful contributions, which unfortunately are impossible to list exhaustively. Special thanks for reading earlier drafts and giving us detailed feedback, go to our second reader, Bob Kowalski, and to Wolfgang Bibel, Elmar Eder, Melvin Fitting, Donald W. Loveland, David Plaisted, and J¨org Siekmann. Work on this chapter started when both of us were members of the Markgraf Karl group at the Universit¨at Kaiserslautern, Germany. With our former colleagues there we had countless fruitful discus- sions, which, again, cannot be credited in detail. During that time this research was supported by the “Sonderforschungsbereich 314, K¨unstliche Intelligenz” of the Deutsche Forschungsgemeinschaft (DFG). -
Journal Abbreviations
Abbreviations of Names of Serials This list gives the form of references used in Mathematical Reviews (MR). not previously listed ⇤ The abbreviation is followed by the complete title, the place of publication journal indexed cover-to-cover § and other pertinent information. † monographic series Update date: July 1, 2016 4OR 4OR. A Quarterly Journal of Operations Research. Springer, Berlin. ISSN Acta Math. Hungar. Acta Mathematica Hungarica. Akad. Kiad´o,Budapest. § 1619-4500. ISSN 0236-5294. 29o Col´oq. Bras. Mat. 29o Col´oquio Brasileiro de Matem´atica. [29th Brazilian Acta Math. Sci. Ser. A Chin. Ed. Acta Mathematica Scientia. Series A. Shuxue † § Mathematics Colloquium] Inst. Nac. Mat. Pura Apl. (IMPA), Rio de Janeiro. Wuli Xuebao. Chinese Edition. Kexue Chubanshe (Science Press), Beijing. ISSN o o † 30 Col´oq. Bras. Mat. 30 Col´oquio Brasileiro de Matem´atica. [30th Brazilian 1003-3998. ⇤ Mathematics Colloquium] Inst. Nac. Mat. Pura Apl. (IMPA), Rio de Janeiro. Acta Math. Sci. Ser. B Engl. Ed. Acta Mathematica Scientia. Series B. English § Edition. Sci. Press Beijing, Beijing. ISSN 0252-9602. † Aastaraam. Eesti Mat. Selts Aastaraamat. Eesti Matemaatika Selts. [Annual. Estonian Mathematical Society] Eesti Mat. Selts, Tartu. ISSN 1406-4316. Acta Math. Sin. (Engl. Ser.) Acta Mathematica Sinica (English Series). § Springer, Berlin. ISSN 1439-8516. † Abel Symp. Abel Symposia. Springer, Heidelberg. ISSN 2193-2808. Abh. Akad. Wiss. G¨ottingen Neue Folge Abhandlungen der Akademie der Acta Math. Sinica (Chin. Ser.) Acta Mathematica Sinica. Chinese Series. † § Wissenschaften zu G¨ottingen. Neue Folge. [Papers of the Academy of Sciences Chinese Math. Soc., Acta Math. Sinica Ed. Comm., Beijing. ISSN 0583-1431. -
The Logic of Brouwer and Heyting
THE LOGIC OF BROUWER AND HEYTING Joan Rand Moschovakis Intuitionistic logic consists of the principles of reasoning which were used in- formally by L. E. J. Brouwer, formalized by A. Heyting (also partially by V. Glivenko), interpreted by A. Kolmogorov, and studied by G. Gentzen and K. G¨odel during the first third of the twentieth century. Formally, intuitionistic first- order predicate logic is a proper subsystem of classical logic, obtained by replacing the law of excluded middle A ∨¬A by ¬A ⊃ (A ⊃ B); it has infinitely many dis- tinct consistent axiomatic extensions, each necessarily contained in classical logic (which is axiomatically complete). However, intuitionistic second-order logic is inconsistent with classical second-order logic. In terms of expressibility, the intu- itionistic logic and language are richer than the classical; as G¨odel and Gentzen showed, classical first-order arithmetic can be faithfully translated into the nega- tive fragment of intuitionistic arithmetic, establishing proof-theoretical equivalence and clarifying the distinction between the classical and constructive consequences of mathematical axioms. The logic of Brouwer and Heyting is effective. The conclusion of an intuitionistic derivation holds with the same degree of constructivity as the premises. Any proof of a disjunction of two statements can be effectively transformed into a proof of one of the disjuncts, while any proof of an existential statement contains an effec- tive prescription for finding a witness. The negation of a statement is interpreted as asserting that the statement is not merely false but absurd, i.e., leads to a contradiction. Brouwer objected to the general law of excluded middle as claim- ing a priori that every mathematical problem has a solution, and to the general law ¬¬A ⊃ A of double negation as asserting that every consistent mathematical statement holds. -
Domain Theory and the Logic of Observable Properties
Domain Theory and the Logic of Observable Properties Samson Abramsky Submitted for the degree of Doctor of Philosophy Queen Mary College University of London October 31st 1987 Abstract The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and systems behaviour developed by Milner, Hennessy et al. based on operational semantics. • Logics of programs. Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of properties of processes). Moreover, the underlying logic is geometric, which can be com- putationally interpreted as the logic of observable properties—i.e. properties which can be determined to hold of a process on the basis of a finite amount of information about its execution. These ideas lead to the following programme: 1. A metalanguage is introduced, comprising • types = universes of discourse for various computational situa- tions. • terms = programs = syntactic intensions for models or points. 2. A standard denotational interpretation of the metalanguage is given, assigning domains to types and domain elements to terms. 3. The metalanguage is also given a logical interpretation, in which types are interpreted as propositional theories and terms are interpreted via a program logic, which axiomatizes the properties they satisfy. 2 4. The two interpretations are related by showing that they are Stone duals of each other. Hence, semantics and logic are guaranteed to be in harmony with each other, and in fact each determines the other up to isomorphism. -
Matematické Ćasopisy Vedené V Databázi SCOPUS
Matematické časopisy vedené v databázi SCOPUS (stav k 21. 1. 2009) Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg Abstract and Applied Analysis ACM Computing Surveys ACM Transactions on Algorithms ACM Transactions on Computational Logic ACM Transactions on Computer Systems ACM Transactions on Speech and Language Processing Acta Applicandae Mathematicae Acta Arithmetica Acta Mathematica Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis Acta Mathematica Hungarica Acta Mathematica Scientia Acta Mathematica Sinica, English Series Acta Mathematica Universitatis Comenianae Acta Mathematicae Applicatae Sinica Acta Mathematicae Applicatae Sinica, English Series Acta Mechanica Sinica Acta Numerica Advanced Nonlinear Studies Advanced Studies in Contemporary Mathematics (Kyungshang) Advanced Studies in Theoretical Physics Advances in Applied Clifford Algebras Advances in Applied Mathematics Advances in Applied Probability Advances in Computational Mathematics Advances in Difference Equations Advances in Geometry Advances in Mathematics Advances in modelling and analysis. A, general mathematical and computer tools Advances in Nonlinear Variational Inequalities Advances in Theoretical and Mathematical Physics Aequationes Mathematicae Algebra and Logic Algebra Colloquium Algebra Universalis Algebras and Representation Theory Algorithms for Molecular Biology American Journal of Mathematical and Management Sciences American Journal of Mathematics American Mathematical Monthly Analysis in Theory and Applications Analysis Mathematica -
UNIVERSALLY BAIRE SETS and GENERIC ABSOLUTENESS TREVOR M. WILSON Introduction Generic Absoluteness Principles Assert That Certai
UNIVERSALLY BAIRE SETS AND GENERIC ABSOLUTENESS TREVOR M. WILSON Abstract. We prove several equivalences and relative consistency re- 2 uBλ sults involving notions of generic absoluteness beyond Woodin's (Σ1) generic absoluteness for a limit of Woodin cardinals λ. In particular,e we R 2 uBλ prove that two-step 9 (Π1) generic absoluteness below a measur- able cardinal that is a limite of Woodin cardinals has high consistency 2 uBλ strength, and that it is equivalent with the existence of trees for (Π1) formulas. The construction of these trees uses a general method for building an absolute complement for a given tree T assuming many \failures of covering" for the models L(T;Vα) below a measurable car- dinal. Introduction Generic absoluteness principles assert that certain properties of the set- theoretic universe cannot be changed by the method of forcing. Some pro- perties, such as the truth or falsity of the Continuum Hypothesis, can always be changed by forcing. Accordingly, one approach to formulating generic ab- soluteness principles is to consider properties of a limited complexity such 1 1 as those corresponding to pointclasses in descriptive set theory: Σ2, Σ3, projective, and so on. (Another approach is to limit the class ofe allowede forcing notions. For a survey of results in this area, see [1].) Shoenfield’s 1 absoluteness theorem implies that Σ2 statements are always generically ab- solute. Generic absoluteness principlese for larger pointclasses tend to be equiconsistent with strong axioms of infinity, and they may also relate to the extent of the universally Baire sets. 1 For example, one-step Σ3 generic absoluteness is shown in [3] to be equiconsistent with the existencee of a Σ2-reflecting cardinal and to be equiv- 1 alent with the statement that every ∆2 set of reals is universally Baire. -
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. -
Martin's Maximum and Weak Square
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000{000 S 0002-9939(XX)0000-0 MARTIN'S MAXIMUM AND WEAK SQUARE JAMES CUMMINGS AND MENACHEM MAGIDOR (Communicated by Julia Knight) Abstract. We analyse the influence of the forcing axiom Martin's Maximum on the existence of square sequences, with a focus on the weak square principle λ,µ. 1. Introduction It is well known that strong forcing axioms (for example PFA, MM) and strong large cardinal axioms (strongly compact, supercompact) exert rather similar kinds of influence on the combinatorics of infinite cardinals. This is perhaps not so sur- prising when we consider the widely held belief that essentially the only way to obtain a model of PFA or MM is to collapse a supercompact cardinal to become !2. We quote a few results which bring out the similarities: 1a) (Solovay [14]) If κ is supercompact then λ fails for all λ ≥ κ. 1b) (Todorˇcevi´c[15]) If PFA holds then λ fails for all λ ≥ !2. ∗ 2a) (Shelah [11]) If κ is supercompact then λ fails for all λ such that cf(λ) < κ < λ. ∗ 2b) (Magidor, see Theorem 1.2) If MM holds then λ fails for all λ such that cf(λ) = !. 3a) (Solovay [13]) If κ is supercompact then SCH holds at all singular cardinals greater than κ. 3b) (Foreman, Magidor and Shelah [7]) If MM holds then SCH holds. 3c) (Viale [16]) If PFA holds then SCH holds. In fact both the p-ideal di- chotomy and the mapping reflection principle (which are well known con- sequences of PFA) suffice to prove SCH. -
E-RECURSIVE INTUITIONS in Memoriam Professor Joseph R. Shoenfield Contents 1. Initial Intuitions 1 2. E-Recursively Enumerable V
E-RECURSIVE INTUITIONS GERALD E. SACKS In Memoriam Professor Joseph R. Shoenfield Abstract. An informal sketch (with intermittent details) of parts of E-Recursion theory, mostly old, some new, that stresses intu- ition. The lack of effective unbounded search is balanced by the availability of divergence witnesses. A set is E-closed iff it is tran- sitive and closed under the application of partial E-recursive func- tions. Some finite injury, forcing, and model theoretic construc- tions can be adapted to E-closed sets that are not 1 admissible. Reflection plays a central role. Contents 1. Initial Intuitions 1 2. E-Recursively Enumerable Versus 1 4 3. E-Recursion Versus -Recursion 6 4. Selection and Reflection 8 5. Finite Injury Arguments and Post’sProblem 12 6. Forcing: C.C.C. Versus Countably Closed 13 7. Model-Theoretic Completeness and Compactness 14 References 16 1. Initial Intuitions One of the central intuitions of classical recursion theory is the effec- tiveness of unbounded search. Let A be a nonempty recursively enu- merable set of nonnegative integers. A member of A can be selected by simply enumerating A until some member appears. This procedure, known as unbounded search, consists of following instructions until a termination point is reached. What eventually appears is not merely Date: December 10, 2010. My thanks to the organizers of "Effective Mathematics of the Uncountable," CUNY Graduate Center, August 2009, for their kind invitation. 1 E-RECURSIVEINTUITIONS 2 some number n A but a computation that reveals n A. Unbounded search in its full2 glory consists of enumerating all computations2 until a suitable computation, if it exists, is found. -
The Development of Mathematical Logic from Russell to Tarski: 1900–1935
The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives . -
The Z of ZF and ZFC and ZF¬C
From SIAM News , Volume 41, Number 1, January/February 2008 The Z of ZF and ZFC and ZF ¬C Ernst Zermelo: An Approach to His Life and Work. By Heinz-Dieter Ebbinghaus, Springer, New York, 2007, 356 pages, $64.95. The Z, of course, is Ernst Zermelo (1871–1953). The F, in case you had forgotten, is Abraham Fraenkel (1891–1965), and the C is the notorious Axiom of Choice. The book under review, by a prominent logician at the University of Freiburg, is a splendid in-depth biography of a complex man, a treatment that shies away neither from personal details nor from the mathematical details of Zermelo’s creations. BOOK R EV IEW Zermelo was a Berliner whose early scientific work was in applied By Philip J. Davis mathematics: His PhD thesis (1894) was in the calculus of variations. He had thought about this topic for at least ten additional years when, in 1904, in collaboration with Hans Hahn, he wrote an article on the subject for the famous Enzyclopedia der Mathematische Wissenschaften . In point of fact, though he is remembered today primarily for his axiomatization of set theory, he never really gave up applications. In his Habilitation thesis (1899), Zermelo was arguing with Ludwig Boltzmann about the foun - dations of the kinetic theory of heat. Around 1929, he was working on the problem of the path of an airplane going in minimal time from A to B at constant speed but against a distribution of winds. This was a problem that engaged Levi-Civita, von Mises, Carathéodory, Philipp Frank, and even more recent investigators.