LOCAL ORDER PROPERTY in NONELEMENTARY CLASSES 1. Introduction in the First Order Case, Victor Harnik and Leo Harrington In
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A Dialogue with Yuri Gurevich About Mathematics, Computer Science and Life ∗
A Dialogue with Yuri Gurevich about Mathematics, Computer Science and Life ∗ Cristian S. Calude, Department of Computer Science The University of Auckland Auckland, New Zealand 1 A Dialogue with Yuri Gurevich about Mathematics, Computer Science and Life Yuri Gurevich is well-known to the readers of this Bulletin. He is a Princi- pal Researcher at Microsoft Research, where he founded a group on Founda- tions of Software Engineering, and a Professor Emeritus at the University of Michigan. His name is most closely associated with abstract state machines but he is known also for his work in logic, complexity theory and software engineering. The Gurevich-Harrington Forgetful Determinacy Theorem is a classical result in game theory. Yuri Gurevich is an ACM Fellow, a Guggen- heim Fellow, and a member of Academia Europaea; he obtained honorary doctorates from Hasselt University in Belgium and Ural State University in Russia. ∗Bulletin of the European Association for Theoretical Computer Science, No. 107, June 2012 1 Cristian Calude: Your background is in mathematics: MSc (1962), PhD (under P. G. Kontorovich, 1964) and Dr of Math (a post-PhD degree in Russia), all at Ural State University. Please reminisce about those years. Yuri Gurevich: I grew up in Chelyabinsk, an industrial city in the Urals, Russia, and was in the first generation of my family to get systematic educa- tion. In 1957, after ten boring years in elementary + middle + high school, I enrolled in the local Polytechnic. I enjoyed student life, but I couldn't draw well, and I hated memorizing things. In the middle of the second year, one math prof advised me to transfer | and wrote a recommendation letter | to the Math Dept of the Ural State University in Ekaterinburg (called Sverdlovsk at the time), about 200 km to the north of Chelyabinsk. -
What Is Mathematics: Gödel's Theorem and Around. by Karlis
1 Version released: January 25, 2015 What is Mathematics: Gödel's Theorem and Around Hyper-textbook for students by Karlis Podnieks, Professor University of Latvia Institute of Mathematics and Computer Science An extended translation of the 2nd edition of my book "Around Gödel's theorem" published in 1992 in Russian (online copy). Diploma, 2000 Diploma, 1999 This work is licensed under a Creative Commons License and is copyrighted © 1997-2015 by me, Karlis Podnieks. This hyper-textbook contains many links to: Wikipedia, the free encyclopedia; MacTutor History of Mathematics archive of the University of St Andrews; MathWorld of Wolfram Research. Are you a platonist? Test yourself. Tuesday, August 26, 1930: Chronology of a turning point in the human intellectua l history... Visiting Gödel in Vienna... An explanation of “The Incomprehensible Effectiveness of Mathematics in the Natural Sciences" (as put by Eugene Wigner). 2 Table of Contents References..........................................................................................................4 1. Platonism, intuition and the nature of mathematics.......................................6 1.1. Platonism – the Philosophy of Working Mathematicians.......................6 1.2. Investigation of Stable Self-contained Models – the True Nature of the Mathematical Method..................................................................................15 1.3. Intuition and Axioms............................................................................20 1.4. Formal Theories....................................................................................27 -
In Memoriam: James Earl Baumgartner (1943-2011)
In memoriam: James Earl Baumgartner (1943–2011) J.A. Larson Department of Mathematics University of Florida, Gainesville Gainesville, FL 32611–8105, USA October 9, 2018 Abstract James Earl Baumgartner (March 23, 1943 – December 28, 2011) came of age mathematically during the emergence of forcing as a fun- damental technique of set theory, and his seminal research changed the way set theory is done. He made fundamental contributions to the development of forcing, to our understanding of uncountable or- ders, to the partition calculus, and to large cardinals and their ideals. He promulgated the use of logic such as absoluteness and elementary submodels to solve problems in set theory, he applied his knowledge of set theory to a variety of areas in collaboration with other math- ematicians, and he encouraged a community of mathematicians with engaging survey talks, enthusiastic discussions of open problems, and friendly mathematical conversations. arXiv:1705.02219v1 [math.HO] 2 May 2017 1 Overview of Baumgartner’s Life James E. Baumgartner was born on March 23, 1943 in Wichita, Kansas. His high school days included tennis, football, and leading roles in school plays. In 1960 he entered the California Institute of Technology, but stayed only two years, moving to the University of California, Berkeley in 1962, in part because it was co-educational. There he met and married his wife Yolanda. He continued his interest in drama and mathematics as an undergraduate, earned his A.B. in mathematics in 1964, and continued study as a graduate 1 student. Baumgartner [9, page 2] dated his interest in set theory to the four week long 1967 UCLA Summer Institute on Axiomatic Set Theory.1 The mathematics for his dissertation was completed in spring 1969, and Baumgartner became a John Wesley Young Instructor at Dartmouth College in fall 1969. -
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. -
Interactions Between Computability Theory and Set Theory by Noah
Interactions between computability theory and set theory by Noah Schweber A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Antonio Montalban, Chair Professor Leo Harrington Professor Wesley Holliday Spring 2016 Interactions between computability theory and set theory Copyright 2016 by Noah Schweber i To Emma ii Contents Contents ii 1 Introduction1 1.1 Higher reverse mathematics ........................... 1 1.2 Computability in generic extensions....................... 3 1.3 Further results .................................. 5 2 Higher reverse mathematics, 1/28 2.1 Introduction.................................... 8 2.2 Reverse mathematics beyond type 1....................... 11 2.3 Separating clopen and open determinacy.................... 25 2.4 Conclusion..................................... 38 3 Higher reverse mathematics, 2/2 41 3 3.1 The strength of RCA0 ............................... 41 3.2 Choice principles ................................. 50 4 Computable structures in generic extensions 54 4.1 Introduction.................................... 54 4.2 Generic reducibility................................ 58 4.3 Generic presentability and !2 .......................... 64 4.4 Generically presentable rigid structures..................... 71 5 Expansions and reducts of R 74 5.1 Introduction.................................... 74 ∗ 5.2 Rexp ≡w R ....................................