Maximal Realcompact Spaces and Measurable Cardinals 1
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Set-Theoretic Geology, the Ultimate Inner Model, and New Axioms
Set-theoretic Geology, the Ultimate Inner Model, and New Axioms Justin William Henry Cavitt (860) 949-5686 [email protected] Advisor: W. Hugh Woodin Harvard University March 20, 2017 Submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Mathematics and Philosophy Contents 1 Introduction 2 1.1 Author’s Note . .4 1.2 Acknowledgements . .4 2 The Independence Problem 5 2.1 Gödelian Independence and Consistency Strength . .5 2.2 Forcing and Natural Independence . .7 2.2.1 Basics of Forcing . .8 2.2.2 Forcing Facts . 11 2.2.3 The Space of All Forcing Extensions: The Generic Multiverse 15 2.3 Recap . 16 3 Approaches to New Axioms 17 3.1 Large Cardinals . 17 3.2 Inner Model Theory . 25 3.2.1 Basic Facts . 26 3.2.2 The Constructible Universe . 30 3.2.3 Other Inner Models . 35 3.2.4 Relative Constructibility . 38 3.3 Recap . 39 4 Ultimate L 40 4.1 The Axiom V = Ultimate L ..................... 41 4.2 Central Features of Ultimate L .................... 42 4.3 Further Philosophical Considerations . 47 4.4 Recap . 51 1 5 Set-theoretic Geology 52 5.1 Preliminaries . 52 5.2 The Downward Directed Grounds Hypothesis . 54 5.2.1 Bukovský’s Theorem . 54 5.2.2 The Main Argument . 61 5.3 Main Results . 65 5.4 Recap . 74 6 Conclusion 74 7 Appendix 75 7.1 Notation . 75 7.2 The ZFC Axioms . 76 7.3 The Ordinals . 77 7.4 The Universe of Sets . 77 7.5 Transitive Models and Absoluteness . -
O&ONSTRUCTIBLE UNIVERSE and MEASURABLE CARDINALS 0
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Annals of Pure and Applied Logic 30 (1986) 293-320 293 North-Holland o&ONSTRUCTIBLE UNIVERSE AND MEASURABLE CARDINALS Claude SURESON Dkpartement de Mathkmatiques, Universitk de Caen, 1403.2 Caen, France Communicated by A. Nerode Received 23 September 1984 In analogy with K. Godel’s model L, C. Chang [l] formulated the wr- constructible universe C”‘, using the infinitary language L,,,, instead of the language of Set Theory L,,. The cumulative hierarchy of sets obtained in this way has many similarities with the hierarchy of the constructible universe (except for a major point: the axiom of choice [l], [9]). C”’ can also be characterized as the least inner model closed under arbitrary countable sequences. This paper is inspired by results of R. Jensen and J. Silver concerning the existence of O# and the covering property for L. We consider here a stronger notion of indiscernibles for the model C”’ and we say that C”’ satisfies the ‘covering property’ if any set of ordinals X in the universe can be covered by a set in C”’ of cardinality ]X]‘O. The existence of ‘indiscernibles’ for C”’ is also linked to large cardinal assumptions, and our main result (in ZFC) can be summarized as shown in Diagram 1: Diagram 1. The first part is devoted to the study of indiscernibles for PI. We prove the implications (1) and (2). In the second section, we deal with the covering property and show (3). -
On Rings of Sets II. Zero-Sets
ON RINGS OF SETS H. ZERO-SETS Dedicated to the memory of Hanna Neumann T. P. SPEED (Received 2 May 1972) Communicated by G. B. Preston Introduction In an earlier paper [llj we discussed the problem of when an (m, n)-complete lattice L is isomorphic to an (m, n)-ring of sets. The condition obtained was simply that there should exist sufficiently many prime ideals of a certain kind, and illustrations were given from topology and elsewhere. However, in these illus- trations the prime ideals in question were all principal, and it is desirable to find and study examples where this simplification does not occur. Such an example is the lattice Z(X) of all zero-sets of a topological space X; we refer to Gillman and Jerison [5] for the simple proof that TAX) is a (2, <r)-ring of subsets of X, where we denote aleph-zero by a. Lattices of the form Z(X) have occurred recently in lattice theory in a number of places, see, for example, Mandelker [10] and Cornish [4]. These writers have used such lattices to provide examples which illuminate a number of results concerning annihilators and Stone lattices. We also note that, following Alexandroff, a construction of the Stone-Cech compactification can be given using ultrafilters on Z(X); the more recent Hewitt realcompactification can be done similarly, and these topics are discussed in [5]. A relation between these two streams of development will be given below. In yet another context, Gordon [6], extending some aspects of the work of Lorch [9], introduced the notion of a zero-set space (X,2£). -
Determinacy and Large Cardinals
Determinacy and Large Cardinals Itay Neeman∗ Abstract. The principle of determinacy has been crucial to the study of definable sets of real numbers. This paper surveys some of the uses of determinacy, concentrating specifically on the connection between determinacy and large cardinals, and takes this connection further, to the level of games of length ω1. Mathematics Subject Classification (2000). 03E55; 03E60; 03E45; 03E15. Keywords. Determinacy, iteration trees, large cardinals, long games, Woodin cardinals. 1. Determinacy Let ωω denote the set of infinite sequences of natural numbers. For A ⊂ ωω let Gω(A) denote the length ω game with payoff A. The format of Gω(A) is displayed in Diagram 1. Two players, denoted I and II, alternate playing natural numbers forming together a sequence x = hx(n) | n < ωi in ωω called a run of the game. The run is won by player I if x ∈ A, and otherwise the run is won by player II. I x(0) x(2) ...... II x(1) x(3) ...... Diagram 1. The game Gω(A). A game is determined if one of the players has a winning strategy. The set A is ω determined if Gω(A) is determined. For Γ ⊂ P(ω ), det(Γ) denotes the statement that all sets in Γ are determined. Using the axiom of choice, or more specifically using a wellordering of the reals, it is easy to construct a non-determined set A. det(P(ωω)) is therefore false. On the other hand it has become clear through research over the years that det(Γ) is true if all the sets in Γ are definable by some concrete means. -
Extension of Continuous Functions Into Uniform Spaces Salvadorhernández
proceedings of the american mathematical society Volume 97, Number 2, June 1986 EXTENSION OF CONTINUOUS FUNCTIONS INTO UNIFORM SPACES SALVADORHERNÁNDEZ ABSTRACT. Let X be a dense subspace of a topological space X, let Y be a uniformizable space, and let /: X —►Y a continuous map. In this paper we study the problem of the existence of a continuous extension of / to the space T. Thus we generalize basic results of Taimanov, Engelking and Blefko- Mrówka on extension of continuous functions. As a consequence, if D is a nest generated intersection ring on X, we obtain a necessary and sufficient condition for the existence of a continuous extension to v(X, D), of a continuous function over X, when the range of the map is a uniformizable space, and we apply this to realcompact spaces. Finally, we suppose each point of T\X has a countable neighbourhood base, and we obtain a generalization of a theorem by Blair, herewith giving a solution to a question proposed by Blair. In the sequel the word space will designate a topological space. By pX we shall denote a uniform space, p being a collection of covers in the sense of Tukey [13] and Isbell [10]. 7/iX will mean the completion of the space pX. Let p be a uniformity on X and let m be an infinite'cardinal number, then pm will denote the uniformity on X generated by the //-covers of power < m (we only consider p and m for which pm is definable; see [6, 10 and 14]). The compact uniform space ipx0X is called the Samuel compactification of pX, denoted by spX. -
The Ultrafilter Characterization of Huge Cardinals 587
proceedings of the american mathematical society Volume 90, Number 4, April 1984 THE ULTRAFILTERCHARACTERIZATION OF HUGE CARDINALS ROBERT J. MIGNONE1 Abstract. A huge cardinal can be characterized using ultrafilters. After an argu- ment is made for a particular ultrafilter characterization, it is used to prove the existence of a measurable cardinal above the huge cardinal, and an ultrafilter over the set of all subsets of this measurable cardinal of size smaller than the huge cardinal. Finally, this last ultrafilter is disassembled intact by a process which often produces a different ultrafilter from the one started out with. An important point of this paper is given the existence of the particular ultrafilter characterization of a huge cardinal mentioned above these results are proved in Zermelo-Fraenkel set theory without the axiom of choice. Introduction. The notion of a huge cardinal was first introduced by Kunen. Its primary definition is given in terms of elementary embeddings. And just as measura- ble and supercompact cardinals have ultrafilter characterizations in ZFC, a huge cardinal can also be characterized by ultrafilters. In §1, after the elementary embedding definition for a huge cardinal and two equivalent ultrafilter characterizations are given, an argument is made for one characterization above the other in instances where the axiom of choice is not available. Once an ultrafilter characterization for k being huge is settled upon—which defines k as huge if for some X > k there exists a K-complete, normal ultrafilter over {x C X: x = k}—two theorems are proved in §2 which state that providing such an ultrafilter exists over {x czX: x = k}, then X is measurable and k is A-supercompact. -
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". -
Bibliography
Bibliography AlexandrotT, P.: 1939 Bikompakte Erweiterungen topologischer Raume. Mat. Sbornik 5, 403-423 (1939). (Russian. German summary.) MR 1, p. 318. AlexandrotT, P. and Hopf, H.: 1935 Topologie, Berlin: Springer 1935. AlexandrotT, P., and Urysohn, P.: 1929 Memoire sur les espaces topologiques compacts. Verh. Akad. Wetensch. Amster dam 14, 1-96 (1929). Alb,R.A.: 1969 Uniformities and embeddings. Proc. Int. Sympos. on Topology and Its Appls. (Herceg-Novi, 1968), pp. 45-59. Belgrad: Savez Drustava Mat. Fiz, i Astronom. 1969. MR 42 # 3737. 1972 Some Tietze type extension theorems. General Topology and Its Relations to Modern Analysis and Algebra ,III (Proc. Third Prague Topological Sympos., 1971), pp. 23-27, Prague: Academia 1972. Alb, R. A., Imler, L., and Shapiro, H. L.: 1970 P- and z-embedded subspaces. Math. Ann. 188, 13-22 (1970). MR 42# 1062. Alb, R. A. and Sennot, L. : 1971 Extending linear space-valued functions. Math. Ann. 191, 79-86 (1971). MR 43#6877. 1972 Collection wise normality and the extension of functions on product spaces. Fund. Math. 76, 231-243 (1972). Alb, R. A. and Shapiro, H. L.: 1968A A note on compactifications and semi-normal spaces. 1. Austral. Math. Soc. 8, 102-108 (1968). MR 37#3527. 1968B Normal bases and compactifications. Math. Ann. 175, 337-340 (1968). MR 36#3312. 1969A Wallman compact and realcompact spaces. Contributions to Extension Theory of Topological Structures (Proc. Sympos., Berlin, 1967), pp. 9-14. BerlIn: Deut scher Verlag Wissensch. 1969. MR 40#872. 1969B ~-realcompactifications and normal bases. 1. Austral. Math. Soc. 9, 489-495 (1969). MR 39#3455. -
Arxiv:Math/0009240V1
A simple maximality principle Joel David Hamkins∗ The City University of New York† Carnegie Mellon University‡ http://math.gc.cuny.edu/faculty/hamkins October 24, 2018 Abstract. In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence ϕ holding in some ˙ forcing extension V P and all subsequent extensions V P∗Q holds already in V . It follows, in fact, that such sentences must also hold in all forcing extensions of V . In modal terms, therefore, the Maximality Principle is expressed by the scheme (✸ ϕ) =⇒ ϕ, and is equivalent to the modal theory S5. In this article, I prove that the Maximality Principle is relatively consistent with zfc. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in ϕ, is equiconsistent with the scheme asserting that Vδ ≺ V for an inaccessible cardinal δ, which in turn is equiconsistent with the scheme asserting that ord is mp mp V Mahlo. The strongest principle along these lines is ∼ , which asserts that ∼ holds in and all forcing extensions. From this, it follows that 0# exists, that x# exists for every set x, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain. 1 The Maximality Principle arXiv:math/0009240v1 [math.LO] 28 Sep 2000 Christophe Chalons has introduced a delightful new axiom, asserting in plain language that anything that is forceable and not subsequently unforceable is true. Specifically, in [2] he proposes the following principle: ∗My research has been supported by grants from the PSC-CUNY Research Foundation and the NSF. -
Algebraic Properties of Certain Rings of Continuous Functions
Pacific Journal of Mathematics ALGEBRAIC PROPERTIES OF CERTAIN RINGS OF CONTINUOUS FUNCTIONS LI PI SU Vol. 27, No. 1 January 1968 PACIFIC JOURNAL OF MATHEMATICS Vol. 27, No. 1, 1968 ALGEBRAIC PROPERTIES OF CERTAIN RINGS OF CONTINUOUS FUNCTIONS Li PI SU n f f Let X and Fbe any subsets of E , and (X , dj and (Y , d2) be any metric spaces. Let O(X), 0 ^ m ^ oo, denote the ring r of m-differentiable functions on X, and Lc(X ) be the ring of the functions which are Lipschitzian on each compact subset of X', and L(Xf) be the ring of the bounded Lipschitzian functions onl'. The relations between algebraic properties m of C (X), (resp. Le(X') or L(X) and the topological properties of X (resp. X') are studied. It is proved that if X and Y, (resp. f 1 (X , dί) and (Y , d2)) are m-realcompact, (resp. Lc-real-compact m or compact) then O(X) = C {Y) (resp. Le(X') = LC(Y>) or L(Xf) = L(Y') if and only if X and Y are O-diffeomorphic 1 (resp. (X ', di) and (Y', d2) are Lc or L-homeomorphic). During the last twenty years, the relations between the algebraic properties of Cm(X) and Cm(Y) and the topological properties of X and Y have been investigated by Hewitt [4], Myers [9], Pursell [11], Nakai [10], and Gillman and Jerison [3], where m is a positive integer, zero or infinite. In 1963, Sherbert [12] studied the ring L(X). Recently, Magill, [6] has obtained the algebraic condition relating C(X) and C(Y) (i. -
Axiomatic Set Theory
http://dx.doi.org/10.1090/pspum/013.1 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume XIII, Part I AXIOMATIC SET THEORY AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 1971 Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society Held at the University of California Los Angeles, California July 10-August 5,1967 Prepared by the American Mathematical Society under National Science Foundation Grant GP-6698 Edited by DANA S. SCOTT A MS 1970 Subject Classifications Primary 02-02,04-02,02K15 Secondary 02B15, 02B25, 02C15, 02F27, 02F29, 02F35, 02H13, 02J05, 02KO5, 02K10, 02K20. 02K25, 02K30, 02K35, 04A10, 04A15, 04A20, 04A25, 04A30, 18A15, 28A05, 28A10 28A60 International Standard Serial Number 0082-0717 International Standard Book Number 0-8218-0245-3 Library of Congress Catalog Number 78-125172 Copyright © 1971 by the American Mathematical Society Printed in the United States of America Reprinted 1987 All rights reserved except those granted to the United States Government May not be reproduced in any form without permission of the publishers The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. @ TABLE OF CONTENTS Foreword v Sets constructible using LKK By C. C. CHANG 1 Comments on the foundations of set theory By PAUL J. COHEN 9 Unsolved problems in set theory By P. ERDOS and A. HAJNAL 17 A more explicit set theory By HARVEY FRIEDMAN 49 Sets, semisets, models By PETR HAJEK 67 The Boolean prime ideal theorem does not imply the axiom of choice By J. D. HALPERN and A. -
Ultrafilters Over a Measurable Cardinal A
Annals of Mathematical Logic ! ! (1976) 315-356 O North-Holland Publishing Company ULTRAFILTERS OVER A MEASURABLE CARDINAL A. KANAMORI * Departmen! of MQ~hematics, [fniverlily of Cahfornia, Be,'k.eley, CA 94720, U.S.A. Received 4 April 1975 0. Introduction The extensive theory that exists on ~ca, the set of uitrafilters ov-.r the integers, suggests an analogous study of the family of g-complete ultrafih~rs over a measurable cardinal g > w. This paper is devoted to such a study, with emphasis on those aspects which make the un- countable case interesting and distinctive. Section 1 is a preliminary section, recapitulating some knov, n con- cepts and results in the theory of ultrafilters, while Section 2 intro- duces the convenient frameworR of Puritz for discussing elementary embeddings of totally ordered structures. Section 3 then begins the study in earnest, and introduces a function r on ultrafilters which is a measure of complexi*7. Section 4 is devoted to p-points; partition properties akin to the familiar Ramsey property of normal ultrafilters are shown to yield non-trivial p-points, and examples are constructed. In Section 5 sum and limit constructions are considered; a new proof of a theorem of $oiovay and a generalization are given, and R is shown that the Rudin-Frolik tree ca,not have much height. Finally, Section 6 discusses filter related formulations of the well-known Jonsson and Rowbottom properties of cardinals. The notation used in this paper is much as in the most recent set theoretical literature, but the following are specified: The letters a,/3, % 6 ..