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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. -
Shorter Curriculum Vitae Akihiro Kanamori
SHORTER CURRICULUM VITAE AKIHIRO KANAMORI DESCRIPTION: Born 23 October 1948 in Tokyo, Japan; now a United States citizen. DEGREES: 1966-1970 California Institute of Technology, Bachelor of Science. 1970-1975 University of Cambridge (King's College), Doctor of Philosophy. Subject: Set Theory, Mathematics. Thesis: Ultrafilters over Uncountable Cardinals. Advisor: A.R.D. Mathias. This involved one year of research at: 1972-1973 University of Wisconsin, Madison. Advisor: K. Kunen. PROFESSIONAL EXPERIENCE: 1975-1977 Lectureship at the University of California, Berkeley. 1977-1981 Benjamin Pierce Assistant Professorship at Harvard University. 1981-1982 Assistant Professorship at Baruch College of the City University of New York. 1982-1983 Visiting Associate Professorship at Boston University. 1983-1992 Associate Professorship at Boston University. 1988-1989 Berman Visiting Professorship, Institute of Mathematics, Hebrew University of Jerusalem. 1992- Professorship at Boston University. 1995 Visiting Professorship, Institute of Mathematics, Hebrew Universiy of Jerusalem. 2002-2003 Senior Fellow of the Dibner Institute for the History of Science and Technology. Visiting Scholar at the Department of the History of Science at Harvard University. 1 2009-2010 Senior Fellow of the Lichtenberg-Kolleg, Institute for Advanced Study, G¨ottingen,Germany. Lecture Course on Set Theory, Mathematische Institut, G¨ottingen,Germany, June-July 2010. FELLOWSHIPS AND AWARDS: Marshall Scholarship (British Government), 1970-1972. Danforth Foundation Fellowship, 1970-1975. Woodrow Wilson Foundation Fellowship, 1970. 1984 New England Open Co-Champion of Chess. Equal First 1986 Greater Boston Chess Open. Equal Second, 1987 Massachusetts Chess Open Championship. Equal Sixth, 1989 Israel Open. Class Prize, 1992 New England Open Championship. 2002-2003 Senior Fellowship, Dibner Institute for the History of Science and Technology. -
Mathematics of the Gateway Arch Page 220
ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com Highlights in Springer’s eBook of the American Mathematical Society Collection February 2010 Volume 57, Number 2 An Invitation to Cauchy-Riemann NEW 4TH NEW NEW EDITION and Sub-Riemannian Geometries 2010. XIX, 294 p. 25 illus. 4th ed. 2010. VIII, 274 p. 250 2010. XII, 475 p. 79 illus., 76 in 2010. XII, 376 p. 8 illus. (Copernicus) Dustjacket illus., 6 in color. Hardcover color. (Undergraduate Texts in (Problem Books in Mathematics) page 208 ISBN 978-1-84882-538-3 ISBN 978-3-642-00855-9 Mathematics) Hardcover Hardcover $27.50 $49.95 ISBN 978-1-4419-1620-4 ISBN 978-0-387-87861-4 $69.95 $69.95 Mathematics of the Gateway Arch page 220 Model Theory and Complex Geometry 2ND page 230 JOURNAL JOURNAL EDITION NEW 2nd ed. 1993. Corr. 3rd printing 2010. XVIII, 326 p. 49 illus. ISSN 1139-1138 (print version) ISSN 0019-5588 (print version) St. Paul Meeting 2010. XVI, 528 p. (Springer Series (Universitext) Softcover ISSN 1988-2807 (electronic Journal No. 13226 in Computational Mathematics, ISBN 978-0-387-09638-4 version) page 315 Volume 8) Softcover $59.95 Journal No. 13163 ISBN 978-3-642-05163-0 Volume 57, Number 2, Pages 201–328, February 2010 $79.95 Albuquerque Meeting page 318 For access check with your librarian Easy Ways to Order for the Americas Write: Springer Order Department, PO Box 2485, Secaucus, NJ 07096-2485, USA Call: (toll free) 1-800-SPRINGER Fax: 1-201-348-4505 Email: [email protected] or for outside the Americas Write: Springer Customer Service Center GmbH, Haberstrasse 7, 69126 Heidelberg, Germany Call: +49 (0) 6221-345-4301 Fax : +49 (0) 6221-345-4229 Email: [email protected] Prices are subject to change without notice. -
A NEW L¨OWENHEIM-SKOLEM THEOREM 1. Introduction The
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 5, Pages 1693–1715 S 0002-9947(04)03445-2 Article electronically published on December 16, 2004 ANEWLOWENHEIM-SKOLEM¨ THEOREM MATTHEW FOREMAN AND STEVO TODORCEVIC Abstract. This paper establishes a refinement of the classical L¨owenheim- Skolem theorem. The main result shows that any first order structure has a countable elementary substructure with strong second order properties. Sev- eral consequences for Singular Cardinals Combinatorics are deduced from this. 1. Introduction The L¨owenheim-Skolem Theorem [21] is one of the classical and formative results of first order logic. Its consequences have been important technically in the devel- opment of Model Theory, Set Theory and interesting from the point of view of the Philosophy of Mathematics [16]. In this paper we improve the L¨owenheim-Skolem Theorem and deduce important combinatorial consequences from the stronger ver- sion. There have been many attempts to improve and generalize properties of first order logic to stronger logics. The literature is much too large to survey here. One property that defines first order logic is the fact that every infinite structure in a countable language has a countable elementary substructure. This is not true of many stronger logics such as second order logic. In particular, in the classical L¨owenheim-Skolem theorem, one has no control over the second order properties of the elementary substructure. In this paper we prove that if one fixes in advance a collection of “intervals” around each point in a particular domain κ (e.g. a club guessing ladder system), then for all real numbers r and all structures A with domain κ, there is a countable elementary substructure of A that has non-empty intersection with exactly those intervals specified by r. -
The Paisley Directory and General Advertiser
H to FOUNDED BY SIR PETER COATS, I87O. REFERENCE DEPARTMENT P.C. xa^^o No Book to be taken out of the Room. ^i -^A 2 347137 21 Digitized by the Internet Arciiive in 2010 witin funding from National Library of Scotland http://www.archive.org/details/paisleydirecto189900unse THE . PAISLEY DIRECTORY AKD GENERAL ADVERTISER FOR 1899-1900. INCLUDING COMPREHENSIVE AND ACCURATE DIRECTORD S RENFREV\^ BLACKSTON, en JOHNSTONE, CLIPPENS, QUARRELTON, LINWOOD, ELDERSLIE, HOWWOOD, m INKERMANN, KILBARCHAN, Etc., >- WITH AN APPENDIX, m Comprising a Copious List of the Public Boards, Institutions, Societies, connected with the various localities. < a. -,.. r PAISLEY : J. & J, COOK, "GAZETTE" BUILDINGS, 94 HIGH STREET. MDCCCXCIX. FC/S30 THE HOLM LAUNDRY, ]P -A. I S X. E "2^ (Telephone 288). THE NEWEST AND BEST IN TOWN. GRASS PARK FOR BLEACHING. FIRST-CLASS CARPET BEATER. G-OODS COIiIiUCTED and DEIiIVERED FREE. GARDNER & CO. Nothing makes HOME so CLEAN, BRIGHT, and ATTRACTIVE as a liberal use of R 1 S©np Powder. Sole Makers : ISPALE & M'CALLUM, PAISLEY. WILLIAM CURRIE, JVIONUMENTAL SeULPiTOR, BROOWILANDS STREET, PAISLEY (OPPOSITE CEMETERY GATE). MONUMENTS, in Granite, Marble, and Freestone executed to any design. MONUMENTS AND HEADSTONES CLEANED AND REPAIRED. INSCRIPTIONS CUT IN EVERY STYLE. JOBBINGS IN TOWN and COUNTRY PUNCTUALLY ATTENDED TO. DESIGNS AND ESTIMATES FREE ON APPLICATION. PREFACE. The Publishers have pleasure ia issuing the present Directory, and hope that it will be found full and accurate. After the General and Trades Directories will be found a copious Appendix, giving the present office - bearers and directorates of the various local public bodies and societies. The Publishers have pleasure in acknowledging the kind- ness of ladies and gentlemen connected with the various associations for supplying information, and also beg to thank their numerous Subscribers and Advertisers for their patronage. -
Of the American Mathematical Society November 2018 Volume 65, Number 10
ISSN 0002-9920 (print) NoticesISSN 1088-9477 (online) of the American Mathematical Society November 2018 Volume 65, Number 10 A Tribute to Maryam Mirzakhani, p. 1221 The Maryam INTRODUCING Mirzakhani Fund for The Next Generation Photo courtesy Stanford University Photo courtesy To commemorate Maryam Mirzakhani, the AMS has created The Maryam Mirzakhani Fund for The Next Generation, an endowment that exclusively supports programs for doctoral and postdoctoral scholars. It will assist rising mathematicians each year at modest but impactful levels, with funding for travel grants, collaboration support, mentoring, and more. A donation to the Maryam Mirzakhani Fund honors her accomplishments by helping emerging mathematicians now and in the future. To give online, go to www.ams.org/ams-donations and select “Maryam Mirzakhani Fund for The Next Generation”. Want to learn more? Visit www.ams.org/giving-mirzakhani THANK YOU AMS Development Offi ce 401.455.4111 [email protected] I like crossing the imaginary boundaries Notices people set up between different fields… —Maryam Mirzakhani of the American Mathematical Society November 2018 FEATURED 1221684 1250 261261 Maryam Mirzakhani: AMS Southeastern Graduate Student Section Sectional Sampler Ryan Hynd Interview 1977–2017 Alexander Diaz-Lopez Coordinating Editors Hélène Barcelo Jonathan D. Hauenstein and Kathryn Mann WHAT IS...a Borel Reduction? and Stephen Kennedy Matthew Foreman In this month of the American Thanksgiving, it seems appropriate to give thanks and honor to Maryam Mirzakhani, who in her short life contributed so greatly to mathematics, our community, and our future. In this issue her colleagues and students kindly share with us her mathematics and her life. -
1 Lessons and Interdisciplinary Projects That Explore Reasons For
Lessons and Interdisciplinary Projects That Explore Reasons for Migration and the Experience of Immigrants to the United States Members of the Hofstra University New Teachers Network teach in New York City and suburban Long Island middle schools and high schools. Most participants in the group work in schools where a significant number of students are immigrants to the United States or the children of immigrants. Because of this, the experience of immigrants to the United States plays a major role in their curricula, especially in social studies and English. They developed the following lessons and projects collaboratively with colleagues in their schools and at monthly network meetings. Michael Pezone and Joseph Spero teach Social Studies and English at Junior High School 190 in Queens, New York. It is one of most ethnically diverse schools in New York City, with students whose families come from India, Pakistan, China, the Middle East, the Philippines, Africa, the former Soviet Union, the Caribbean, and Latin America. Even the students who identify themselves as American usually have ancestors who migrated to the United States in the early twentieth century. Because of the nature of the student body, and their belief that young people should be able to see themselves in history and literature, Michael and Joseph make an examination of the immigrant experience in the United States a central feature of the curriculum. Projects in their classes include student presentations of artifacts that their families retain from their original cultures. Over the years, students have presented and discussed articles of clothing, money and stamps, eating and cooking utensils, religious paraphernalia, photographs, and different types of food. -
Potent Axioms by Matthew Foreman
TRANSACTIONS of the AMERICAN MATHEMATICAL SOCIETY Volume 294, Number 1, March 1986 POTENT AXIOMS BY MATTHEW FOREMAN Abstract. This paper suggests alternatives to the ordinary large cardinal axioms of set theory. These axioms can be viewed as generalizations of large cardinals and exhibit many of the same phenomena. They are shown to imply the G.C.H., every set of reals in L(R) is Lebesgue measurable, and various results in combinatorics, algebra and model theory. Potent axioms. A great number of mathematical conjectures have been shown to be independent of the ordinary axioms of Zermelo-Frankel set theory with the axiom of choice. They include propositions from virtually every area of mathematics, including analysis [W], algebra [Sh2], topology and combinatorics. Apparently the only way of settling these questions is to strengthen ZFC with natural axioms that answer these questions. Two major axiom systems have been proposed; determinacy axioms and large cardinal axioms. They have been successful to a large extent but have not answered many crucial questions such as the continuum hypothesis. This paper proposes a line of axioms that can be viewed as generalizations of large cardinal axioms. It is shown that they settle the G.CH. (positively) as well as several classical questions in set theory. It is shown that they form a strict hierarchy in consistency strength, and a hueristic argument is given to show that they may be stronger in consistency strength than ordinary large cardinal axioms. The author conjectures that they imply de- terminacy axioms. Large cardinal axioms generally are equivalent to the existence of monomor- phisms j from V (the mathematical universe) into a transitive class M. -
Açık Ihale Usulü Ile Ihale Edilen 2.525 Adet Yabancı Dilde Kitap Mal Alımı Idari Şartnamesi I
Açık ihale usulü ile ihale edilen 2.525 Adet Yabancı Dilde Kitap mal alımı idari şartnamesi I - İHALENİN KONUSU VE TEKLİF VERMEYE İLİŞKİN HUSUSLAR Madde 1 - İdareye ilişkin bilgiler 1.1. İdarenin; a) Adı: ORTA DOĞU TEKNİK ÜNİVERSİTESİ İdari ve Mali İşler Daire Başkanlığı b) Adresi: INÖNÜ BULVARI ODTÜ YERLESKESI MÜHENDISLIK MERKEZ BINASI KAT:1 112 06531 ÇANKAYA ANKARA c) Telefon numarası: 3122103845 ç) Faks numarası: 3122103731 d) Elektronik posta adresi: [email protected] e) İlgili personelinin adı, soyadı ve unvanı: İdari Personel: Ayfer BAYRAM - İç Satınalma Müdürü Tel: 0312 210 3845 Teknik Personel: Recep GEYİKÇİ - ODTÜ Kütüphane ve Dokümantasyon Daire Bşk. Şube Müdürü Tel: 0312 210 2788 1.2. İstekliler, ihaleye ilişkin bilgileri yukarıdaki adres ve numaralardan görevli personelle irtibat kurmak suretiyle temin edebilirler. Madde 2 - İhale konusu alıma ilişkin bilgiler 2.1. İhale konusu malın; a) Adı: 2.525 Adet Yabancı Dilde Kitap alımı b) Varsa kodu: c) Miktarı ve türü: Malın miktarı ve türü ekte yer almaktadır. EĞİTİM : 363 ADET FEN EDEBİYAT : 980 ADET İKTİSADİ ve İDARİ BİLİMLER : 467 ADET MÜHENDİSLİK : 342 ADET SANAT ve MİMARİ : 373 ADET TOPLAM : 2.525 ADET ç) Teslim edileceği yer: ODTÜ Kütüphane ve Dokümantasyon Daire Başkanlığı d) Bu bent boş bırakılmıştır. Madde 3 - İhaleye ilişkin bilgiler ile ihale ve son teklif verme tarih ve saati 3.1. a) İhale kayıt numarası: 2011/6755 b) İhale usulü: Açık ihale. c) Tekliflerin sunulacağı adres: ODTÜ - İdari ve Mali İşler daire Başkanlığı, İnönü Bulvarı Merkez Mühendislik Binası Kat:1 No: 112 ODTÜ/ANKARA ç) İhalenin yapılacağı adres: ODTÜ - İdari ve Mali İşler daire Başkanlığı, İnönü Bulvarı Merkez Mühendislik Binası Kat:1 No: 112 ODTÜ/ANKARA d) İhale (son teklif verme) tarihi: 28.02.2011 e) İhale (son teklif verme) saati: 11:00 f) İhale komisyonunun toplantı yeri: ODTÜ - İdari ve Mali İşler daire Başkanlığı, İnönü Bulvarı Merkez Mühendislik Binası Kat:1 No: 112 ODTÜ/ANKARA 3.2. -
RECURSION THEORY PROCEEDINGS of SYMPOSIA in PURE MATHEMATICS Volume 42
http://dx.doi.org/10.1090/pspum/042 RECURSION THEORY PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume 42 RECURSION THEORY AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 42 PROCEEDINGS OF THE AMS-ASL SUMMER INSTITUTE ON RECURSION THEORY HELD AT CORNELL UNIVERSITY ITHACA, NEW YORK JUNE 28-JULY 16, 1982 EDITED BY ANIL NERODE AND RICHARD A. SHORE Prepared by the American Mathematical Society with partial support from National Science Foundation grant MCS 8120074 1980 Mathematics Subject Classification. Primary 03D25, 03D30, 03D45, 03D55, 03D60, 03D70, 03D80, 03E05, 03E10, 03E35, 03E45, 03E47, 03F30, 03G05, 13E05, 14M05, 28A12, 90D05, 90D13. Library of Congress Cataloging in Publication Data Main entry under title: Recursion theory. (Proceedings of symposia in pure mathematics; v. 42) Papers presented at the 1982 AMS Summer Research Institute held at Cornell University from June 28 to July 16, 1982, and co-sponsored by the ASL. Bibliography: p. 1. Recursion theory—Addresses, essays, lectures. I. Nerode, Anil. 1932— II. Shore, Richard A., 1946- . III. American Mathematical Society. IV. AMS Summer Research Institute (1982: Cornell University) V. Series. QA9.6.R4 1984 511.3 8448525 ISBN 0-8218-1447-8 COPYING AND REPRINTING. Individual readers of this publication, and nonprofit librar• ies acting for them are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews provided the customary acknowledgement of the sources is given. Republication, systematic copying, or multiple reproduction of any material in this pub• lication (including abstracts) is permitted only under license from the American Mathematical Society. -
Fowler's Commercial Directory of the Principal Towns and Villages in The
Sio-sOT MiStll "Mx*? JSriBrarg & 4iliht0.eum, FOUNDED BY SIR PETER GOATS, I87O. REFERENCE DEPARTMENT P.C. 05 No Book to be taken out of the Room. A^ i> ^ IOK.PI1SI 2 223129 21 Digitized by the Internet Archive in 2010 with funding from National Library of Scotland http://www.archive.org/details/fowlerscommerc184142unse 1 £513 .21 ) FOWLER'S PAISLEY AND JOHNSTONE COMMERCIAL DIRECTORY 1841-42, CONTAIMNG COMPREHENSIVE AND ACCURATE DIRECTORIES OF u ELDERSLIE, PAISLEY, JOHNSTONE, AND LINWOOD, QUARRELTON, AN ALPHABETICAL LIST OF THE Mtxtljmxte, Craters, Mmufattuvtm PRINCIPAL INHABITANTS. Also, a Copious Street Guide of Paisley: AND AN APPENDIX CONTAINING MANY USEFUL LISTS AND TABLES. ALSO, AN ORIGINAL LIST OF THE TRADERS OF PAISLEY IN 1783. TENTH PUBLICATION. PAISLEY: PUBLISHED AND SOLD BY O. FOWLER, BOOKSELLER. {See next page J. NEILSON, PRINTER, PAISLEY. TENTH EDITION OF FOWLER'S DIRECTORY IS MOST RESPECTFULLY DEDICATED TO P. M. STEWART, ESQ., M.P. FOR RENFREWSHIRE ; ARCHIBALD HA ST IE, ESQ., M.P. FOR PAISLEY R. WALLACE, ESQ. OF KELLY, M.P. FOR GREENOCK; SIR JOHN MAXWELL OF POLLOK, BART.; AND THE MAGISTRATES AND TOWN COUNCIL OF PAISLEY, BY THEIR MOST OBEDIENT HUMBLE SERVANT, GEORGE FOWLER. ADDRESS. G. FOWLER, in presenting to the public the Tenth Edition of the Directory, returns his sincere thanks to those who have supported the "Work since its commencement. The Publisher has now been long aware, that in accuracy of de- tails and distinctness of arrangement the chief value of such a work consists ; for these reasons, he has in this, as in his former publications of the Work, both for the Town and County, spared neither labour nor expense in securing these. -
A BRIEF HISTORY of DETERMINACY §1. Introduction
0001 0002 0003 A BRIEF HISTORY OF DETERMINACY 0004 0005 0006 PAUL B. LARSON 0007 0008 0009 0010 x1. Introduction. Determinacy axioms are statements to the effect that 0011 certain games are determined, in that each player in the game has an optimal 0012 strategy. The commonly accepted axioms for mathematics, the Zermelo{ 0013 Fraenkel axioms with the Axiom of Choice (ZFC; see [Jec03, Kun83]), imply 0014 the determinacy of many games that people actually play. This applies in 0015 particular to many games of perfect information, games in which the 0016 players alternate moves which are known to both players, and the outcome 0017 of the game depends only on this list of moves, and not on chance or other 0018 external factors. Games of perfect information which must end in finitely 0019 many moves are determined. This follows from the work of Ernst Zermelo 0020 [Zer13], D´enesK}onig[K}on27]and L´aszl´oK´almar[Kal1928{29], and also 0021 from the independent work of John von Neumann and Oskar Morgenstern 0022 (in their 1944 book, reprinted as [vNM04]). 0023 As pointed out by Stanis law Ulam [Ula60], determinacy for games of 0024 perfect information of a fixed finite length is essentially a theorem of logic. 0025 If we let x1,y1,x2,y2,::: ,xn,yn be variables standing for the moves made by 0026 players player I (who plays x1,::: ,xn) and player II (who plays y1,::: ,yn), 0027 and A (consisting of sequences of length 2n) is the set of runs of the game 0028 for which player I wins, the statement 0029 0030 (1) 9x18y1 ::: 9xn8ynhx1; y1; : : : ; xn; yni 2 A 0031 essentially asserts that the first player has a winning strategy in the game, 0032 and its negation, 0033 0034 (2) 8x19y1 ::: 8xn9ynhx1; y1; : : : ; xn; yni 62 A 0035 essentially asserts that the second player has a winning strategy.1 0036 0037 The author is supported in part by NSF grant DMS-0801009.