Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies (Lecture Notes in Logic, Series Number 36)

"This collection of papers from various areas of mathematical logic showcases the remarkable breadth and richness of the field. Leading authors reveal how contemporary technical results touch upon foundational questions about the nature of mathematics. Highlights of the volume include: a history of Tennenbaum's theorem in arithmetic; a number of papers on Tennenbaum phenomena in weak arithmetics as well as on other aspects of arithmetics, such as interpretability; the transcript of Gödel's previously unpublished 1972-1975 conversations with Sue Toledo, along with an appreciation of the same by Curtis Franks; Hugh Woodin's paper arguing against the generic multiverse view; Anne Troelstra's history of intuitionism through 1991; and Aki Kanamori's history of the Suslin problem in set theory. The book provides a historical and philosophical treatment of particular theorems in arithmetic and set theory, and is ideal for researchers and graduate students in mathematical logic and philosophy of mathematics"-- "The papers collected here engage each of these questions through the veil of particular technical results. For example, the new proof of the irrationality of the square root of two, given by Stanley Tennenbaum in the 1960s and included here, brings into relief questions about the role simplicity plays in our grasp of mathematical proofs. In 1900 Hilbert asked a question which was not given at the Paris conference but which has been recently found in his notes for the list: find a criterion of simplicity in mathematics. The Tennenbaum proof is a particularly striking example of the phenomenon Hilbert contemplated in his 24th Problem"-- Cover......Page 1 Title......Page 5 Copyright......Page 6 Dedication......Page 7 Contents......Page 9 Introduction......Page 11 §1. Suslin’s problem.......Page 17 §2. Consistency of ¬SH.......Page 20 §3. Consistency of SH.......Page 21 §4. Envoi.......Page 25 REFERENCES......Page 26 §1. A tale of two problems......Page 29 §2. The generic-multiverse of sets.......Page 30 §3. Ω-log......Page 35 §4. The Ω conjecture.......Page 37 §5. The complexity of Ω-logic.......Page 39 §6. The weak multiverse laws and H(c+).......Page 40 §7. Conclusions.......Page 42 §8. Appendix.......Page 47 REFERENCES......Page 58 §1. Introduction.......Page 59 §2. Preliminaries.......Page 61 §3. Building ω-models.......Page 65 §4. Models with special properties.......Page 72 §5. ZFfin and PA are not bi-interpretable.......Page 77 §6. Concluding remarks and open questions.......Page 79 REFERENCES......Page 80 §1. Some historical background.......Page 82 §2. Tennenbaum’s theorem.......Page 84 §3. Diophantine problems.......Page 89 REFERENCES......Page 93 §1. Introduction.......Page 96 §2. Preliminaries.......Page 98 3.1. Proof of Theorem A......Page 100 3.2. Proof of Theorem B.......Page 105 3.3. Proofs of Theorems C and D.......Page 106 REFERENCES......Page 108 Background.......Page 109 Wilkie’s theorems and the models of Berarducci and Otero.......Page 110 §1. Axioms for DOI.......Page 112 §2. Diophantine correct rings of Puiseux polynomials......Page 116 Special sequences of polynomials.......Page 120 Theorems on generalized polynomials.......Page 122 §4. A Class of Diophantine correct ordered rings.......Page 123 REFERENCES......Page 127 Tennenbaum’s theorem and recursive reducts......Page 128 §0. Conventions.......Page 130 §1. Rich theories.......Page 131 §2. Thin theories.......Page 134 §3. Examples.......Page 138 §4. Some 1-thin theories.......Page 145 §5. More about LO.......Page 155 REFERENCES......Page 163 §1. Introduction.......Page 166 2.1. Finitist mathematics.......Page 167 2.2. Actualism.......Page 168 3.1. Poincaré.......Page 169 3.2. The semi-intuitionists.......Page 170 3.3. Borel and the continuum.......Page 171 3.4. Weyl.......Page 172 4.1. Early period.......Page 173 4.2. Weak counterexamples and the creative subject.......Page 174 4.3. Brouwer’s programme.......Page 175 5.1. L. E. J. Brouwer and intuitionistic logic.......Page 176 5.3. Formal intuitionistic logic and arithmetic through 1940.......Page 177 5.4. Metamathematics of intuitionistic logic and arithmetic after 1940.......Page 179 5.5. Formulas-as-types.......Page 181 6.1. Choice sequences in Brouwer’s writings.......Page 182 6.2. Axiomatization of intuitionistic analysis.......Page 184 7.1. Classical recursive mathematics.......Page 186 7.2. Constructive recursive mathematics.......Page 188 8.1. Bishop’s constructive mathematics.......Page 189 8.2. The relation of BCM to INT and CRM.......Page 190 §9. Concluding remarks.......Page 191 REFERENCES......Page 192 §1. Introduction:......Page 196 2.1. Murios.......Page 198 2.2. Apeirōn.......Page 202 §3. Recent history of UF.......Page 209 REFERENCES......Page 213 Husserl.Conversation March 3, 1972.......Page 216 Plato paper.......Page 219 Conversation August 21, 1974.......Page 220 Conversation Phone Nov 1.......Page 221 Gödel on Intuitionism.......Page 222 Stanley Tennenbaum’s Socrates......Page 224 REFERENCES......Page 241 Tennenbaum’s proof of the irrationality of v2......Page 242 Machine generated contents note: 1. Introduction Juliette Kennedy and Roman Kossak; 2. Historical remarks on Suslin's problem Akihiro Kanamori; 3. The continuum hypothesis, the generic-multiverse of sets, and the [OMEGA] conjecture W. Hugh Woodin; 4. [omega]-Models of finite set theory Ali Enayat, James H. Schmerl and Albert Visser; 5. Tennenbaum's theorem for models of arithmetic Richard Kaye; 6. Hierarchies of subsystems of weak arithmetic Shahram Mohsenipour; 7. Diophantine correct open induction Sidney Raffer; 8. Tennenbaum's theorem and recursive reducts James H. Schmerl; 9. History of constructivism in the 20th century A. S. Troelstra; 10. A very short history of ultrafinitism Rose M. Cherubin and Mirco A. Mannucci; 11. Sue Toledo's notes of her conversations with Gödel in 1972-1975 Sue Toledo; 12. Stanley Tennenbaum's Socrates Curtis Franks; 13. Tennenbaum's proof of the irrationality of [the square root of] 2́. "The papers collected here engage each of these questions through the veil of particular technical results. For example, the new proof of the irrationality of the square root of two, given by Stanley Tennenbaum in the 1960s and included here, brings into relief questions about the role simplicity plays in our grasp of mathematical proofs. In 1900 Hilbert asked a question which was not given at the Paris conference but which has been recently found in his notes for the list: find a criterion of simplicity in mathematics. The Tennenbaum proof is a particularly striking example of the phenomenon Hilbert contemplated in his 24th Problem"-- Provided by publisher A collection of remarkable papers from various areas of mathematical logic, written by outstanding members of the field