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About Logic and Logicians A palimpsest of essays by Georg Kreisel Selected and arranged by Piergiorgio Odifreddi Volume 2: Mathematics Lógica no Avião ABOUT LOGIC AND LOGICIANS A palimpsest of essays by Georg KREISEL Selected and arranged by Piergiorgio ODIFREDDI Volume II. Mathematics Editorial Board Fernando Ferreira Departamento de Matem´atica Universidade de Lisboa Francisco Miraglia Departamento de Matem´atica Universidade de S~aoPaulo Graham Priest Department of Philosophy The City University of New York Johan van Benthem Department of Philosophy Stanford University Tsinghua University University of Amsterdam Matteo Viale Dipartimento di Matematica \Giuseppe Peano" Universit`adi Torino Piergiorgio Odifreddi, About Logic and Logicians, Volume 2. Bras´ılia: L´ogicano Avi~ao,2019. S´erieA, Volume 2 I.S.B.N. 978-65-900390-1-9 Prefixo Editorial 900390 Obra publicada com o apoio do PPGFIL/UnB. Editor's Preface These books are a first version of Odifreddi's collection of Kreisel's expository papers, which together constitute an extensive, scholarly account of the philo- sophical and mathematical development of many of the most important figures of modern logic; some of those papers are published here for the first time. Odifreddi and Kreisel worked together on these books for several years, and they are the product of long discussions. They finally decided that they would collect those essays of a more expository nature, such as the biographical memoirs of the fellows of the Royal Society (of which Kreisel himself was a member) and other related works. Also included are lecture notes that Kreisel distributed in his classes, such as the first essay printed here, which is on the philosophy of mathematics and geometry. Kreisel himself wrote all the texts, but Odifreddi has made some substantial editorial interventions, rearranging some of the material, breaking the text into sections and paragraphs, inserting titles, moving or removing some notes, and eliminating some digressions. These interventions were made in order to give the essays some of their original freshness and linearity, qualities that were lost in later versions. Some other minor modifications were made here and there, consisting basi- cally in the correction of a small number of erroneous references in the original manuscripts. Kreisel's expository works are invaluable to logicians, and we hope that the reader may find the present edition to his advantage, even if there is still some editorial work to do. Rodrigo Freire. Bras´ılia,April 2019. Contents I ON BROUWER 1 1 BROUWER'S FOUNDATIONS 2 Intuitionistic notions . .3 Brouwer's foundational critique . .5 The famous dispute between Brouwer and Hilbert . .8 Mysticism . 10 2 LUITZEN BROUWER 11 2.1 General Development ....................... 11 Life, family, general interests . 11 Education and academic career . 12 Constructivity . 12 Topology . 13 Intuitionism and formalism . 14 Development of constructive mathematics . 14 Brouwer's controversy with Hilbert . 15 The half-century mark . 16 Solipsism . 17 2.2 Foundations of Mathematics ................... 18 Constructivity on an elementary level . 19 Constructive logic . 21 Infinite proofs . 23 Hilbert's formalist foundations . 25 Ideal mathematician . 26 Looking back . 28 Is mathematics about our own constructions or is it about an ex- ternal reality? . 28 3 BROUWER'S `CAMBRIDGE LECTURES ON INTUITION- ISM' 30 CONTENTS 4 SOME CORRESPONDING HIGH SPOTS OF EARLY CLAS- SICAL AND INTUITIONISTIC LOGIC 33 Completeness with respect to intended meanings . 33 Some early concocted meanings . 35 Philosophical assessment of intended and concocted meanings . 37 Warnings against uses for history or methodology . 40 II ON GODEL¨ 43 5 GODEL'S¨ ELECTION TO THE ROYAL SOCIETY 44 Statement for the Royal Society . 44 G¨odel'sreply . 46 Further developments . 47 Warnings . 47 6 KURT GODEL¨ 49 Sources . 51 6.1 Life and Career ........................... 52 Family background . 52 Growing up in Br¨unnand Brno (1906{1923) . 52 Vienna, with two interludes at Princeton (1923{1938) . 53 Breaking the Austrian connection (1938{39) . 56 The New World: the first 30 years (1939{69) . 57 The final years (1969{1978) . 60 6.2 G¨odel'sFirst Results in Focus ................. 61 Historical perspective . 61 Philosophical perspective . 62 Accentuating the positive: purity of methods . 64 6.3 Background to [1931]: Axiomatization and Formalization 65 From non-elementary axiomatizations to formalizations . 67 Methodenreinheit: how to test philosophical ideals . 68 6.4 The Incompleteness Theorems ................. 69 Formalization and numerical computation: generalities . 69 Incompleteness of formal systems for number theory and beyond . 71 Some lessons from the first incompleteness theorem . 74 Consistency and consistency proofs . 75 Some lessons from the second incompleteness theorem . 78 6.5 Background to [1930]: Elementary Logic in the Twenties 79 Non-categoricity of elementary axioms . 80 Two formulations of completeness (for logical validity) . 81 Herbrand's Theorem . 82 6.6 The Completeness Theorem ................... 82 CONTENTS Enter G¨odel . 83 The Finiteness Theorem . 83 Some lessons from the completeness theorem . 84 6.7 Foundational Bearing of G¨odel'sFirst Results ....... 87 Russell's scheme . 87 Hilbert's scheme . 88 Brouwer's scheme . 90 From foundations to technology . 93 6.8 Background to [1938]: Zermelo's Set Theory ........ 93 Sets before the twenties . 94 Fat hierarchies of sets . 94 Non-elementary axiomatizations . 95 Digression on the passage to formalizations of set theory . 96 Consequences of the non-elementary axioms . 98 Bibliographical remarks . 99 6.9 Constructible Sets ......................... 101 First step: a restricted power set . 101 Second step: the number of iterations . 102 Constructible sets: reculer pour mieux sauter . 102 Bibliographical remarks . 104 GCH: a variant of the axiom of reducibility . 105 Other properties of L . 106 Formal independence results . 108 Some logical and foundational lessons . 109 6.10 G¨odel'sProgram: Axioms of Infinity ............. 112 Enriching the language of formal set theories . 112 Axioms of infinity . 113 Axioms of determinacy . 115 6.11 G¨odel'sfoundational views: balancing the account ..... 118 Successes: mixing the realist and idealist traditions . 118 Neglected problems: beyond na¨ıve idealism . 119 Bibliographical remarks . 121 6.12 Philosophy: Speculations and Reflection ........... 123 General Theory of Relativity . 124 Non-mechanical laws of nature . 125 Chemical evolution of living organisms on earth . 128 General interests (and a contrast) . 128 6.13 Foundations and the Common Understanding ....... 129 7 GODEL'S¨ EARLY WORKS 132 The Pythagorean thesis . 132 Summary . 133 7.1 Background: Doing Sums Formally .............. 134 CONTENTS Numerical equations . 134 Numerical inequalities . 136 Diophantine questions . 136 Thinking about sums: facts of experience . 137 7.2 Two Twists by G¨odelon Cantor's Results .......... 137 A formal counterpart to Cantor's coding arguments . 137 A formal counterpart to Cantor's diagonal arguments . 139 Interpretations . 141 Discussion . 143 Digression on representations (optional) . 144 Systems that prove their own consistency . 148 7.3 Back to the Two Questions ................... 149 Formal (or, equivalently, mechanical) aspects of mathematics: what are they, and what are they supposed to do? . 149 The incompleteness theorem: a literal refutation . 152 The completeness theorem: a Pyrrhic victory . 153 Sensationalism and utilitarianism . 154 Shifts of emphasis: what more do we know from formalization? . 154 What is so wonderful about formalization? . 155 Short answers to the initial questions . 157 Refutations: For a better quality of life . 158 8 GODEL'S¨ LATER WORKS 161 Metamathematics . 162 Logic chopping . 162 8.1 How Adequate Are Those Would-Be Fundamental Meta- mathematical Notions? ([1938], [1939], [1940]) ....... 163 Digression on the commerce of ideas (optional) . 164 8.2 Logic Chopping: Elementary Samples ([1944]) ....... 165 8.3 Absolutes: a Top Priority in the Logical Tradition ([1946]) 170 Autobiographical remarks: absoluteness scaled down . 172 8.4 Selected Thoughts About Sets ................. 173 G¨odel'sown perspective and our agenda . 174 Terminology . 175 Some home truths, half truths and untruths . 175 Axiom of choice . 176 Comprehension . 177 Replacement and (uncountable) strongly inaccessible cardinals . 178 Addendum on replacement (optional) . 179 8.5 Intuitionistic Logic: Hitting a (Little) Moon First, and Then Dreaming of the Stars ................... 181 Background . 181 G¨odel'sscheme . 182 CONTENTS A pyrrhic success . 183 8.6 Cosmology and Some (Even) More Ethereal Ologies ([1949], [1949a], [1950]) ........................... 184 Analysis of language . 185 Sundry tit-bits: old and new . 186 Other ethereal ologies . 187 Digression on logical aspects of theology (for intellectually cheerful readers) . 189 8.7 What Was Lacking (60, 40, or Even 20 Years Ago)? .... 191 8.8 Appendix. A View of Non-Standard Analysis ([1974]) .. 193 The tree (of knowledge) of numbers . 194 Ordering theorems by logical implication . 194 Concrete numerical problems . 195 Disclaimer . 196 A report on impressions of non-standard.
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