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 1.5. Hilbert's Program..................................................................................30 1.6. Some Replies to Critics.........................................................................32 2. Axiomatic Set Theory...................................................................................36 2.1. The Origin of Cantor's Set Theory........................................................36 2.2 Formalization of Cantor's Inconsistent Set Theory................................42 2.3. Zermelo-Fraenkel Axioms....................................................................47 2.4. Around the Continuum Problem...........................................................65 2.4.1. Counting Infinite Sets........................................................................65 2.4.2. Axiom of Constructibility..................................................................74 2.4.3. Axiom of Determinacy.......................................................................77 2.4.4. Large Cardinal Axioms......................................................................80 2.4.5. Ackermann's Set Theory....................................................................89 3. First Order Arithmetic..................................................................................93 3.1. From Peano Axioms to First Order Arithmetic.....................................93 3.2. How to Find Arithmetic in Other Formal Theories............................106 3.3. Representation Theorem.....................................................................110 4. Hilbert's Tenth Problem..............................................................................122 4.1. History of the Problem. Story of the Solution....................................122 4.2. Plan of the Proof.................................................................................135 4.3. Investigation of Fermat's Equation.....................................................138 4.4. Diophantine Representation of Solutions of Fermat's Equation.........144 4.5. Diophantine Representation of the Exponential Function..................148 4.6. Diophantine Representation of Binomial Coefficients and the Factorial Function.....................................................................................................150 4.7. Elimination of Restricted Universal Quantifiers................................153 4.8. 30 Ans Apres.......................................................................................158 5. Incompleteness Theorems..........................................................................160 5.1. Liar's Paradox.....................................................................................160 5.2. Arithmetization and Self-Reference Lemma......................................162 5.3. Gödel's Incompleteness Theorem.......................................................166 5.4. Gödel's Second Incompleteness Theorem..........................................179 6. Around Gödel's Theorem............................................................................188 6.1. Methodological Consequences...........................................................188 6.2. Double Incompleteness Theorem.......................................................193 6.3. Is Mathematics "Creative"?................................................................197 3 6.4. On the Length of Proofs......................................................................201 6.5. Diophantine Incompleteness Theorem: Natural Number System Evolving?...................................................................................................205 6.6. Löb's Theorem....................................................................................208 6.7. Consistent Universal Statements Are Provable..................................209 6.8. Berry's Paradox and Incompleteness. Chaitin's Theorem...................214 Appendix 1. About Model Theory..................................................................222 Appendix 2. Around Ramsey's Theorem........................................................230 Appendix 3. Elements of Category Theory (under construction)..................242 4 References Barwise J. (1942-2000) [1977] Handbook of Mathematical Logic, Elsevier Science Ltd., 1977, 1166 pp. (Russian translation available) Brouwer L. E. J. [1912] Intuitionism and formalism. Inaugural address at the University of Amsterdam (October, 14, 1912). Published in Bull. Amer. Math. Soc., 1913, vol. 20, pp.81-96 (online copy). van Dantzig D. [1955] Is 10**10**10 a finite number? "Dialectica", 1955, vol.9, N 3/4, pp. 273-278 Detlovs V., Podnieks K. [2000] Introduction to Mathematical Logic. Hyper-textbook for students, 2000-2013. Available online. Devlin K. J. [1977] The axiom of constructibility. A guide for the mathematician. "Lecture notes in mathematics", vol. 617, Springer-Verlag, Berlin – Heidelberg – New York, 1977, 96 pp. Feferman S., Friedman H. M., Maddy P., Steel J. R. [2000] Does mathematics need new axioms? "The Bulletin of Symbolic Logic", 2000, vol.6, N 4, pp. 401-446 (see online copy at www.math.ucla.edu/~asl/bsl/0604/0604-001.ps). Hadamard J. [1945] An essay on the psychology of invention in the mathematical field. Princeton, 1945, 143 pp. (Russian translation available) Heijenoort van J. [1967] From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press, 1967, 680 pp. Hersh R. [1979] Some Proposals for Reviving the Philosophy of Mathematics. "Advances in Mathematics", 1979, vol. 31, pp. 31-50. Jech T. Lectures in set theory with particular emphasis on the method of forcing. "Lecture Notes in Mathematics", vol. 217, Springer-Verlag, Berlin – 5 Heidelberg – New York, 1971 (Russian translation available) Keldysh L. V. [1974] The ideas of N.N.Luzin in descriptive set theory. Russian Mathematical Surveys, 1974, vol. 29, N 5, pp. 179-193 (Russian original: Uspekhi matematicheskih nauk, 1974, vol. 29, N 5, pp. 183-196). Mendelson E. [1997] Introduction to Mathematical Logic. Fourth Edition. International Thomson Publishing, 1997, 440 pp. (Russian translation available) Parikh R. [1971] Existence and Feasibility in Arithmetic. JSL, 1971, Vol.36, N 3, pp.494-508. Podnieks K. M. [1975] The double-incompleteness theorem. Scientific Proceedings of Latvia State University, 1975, Vol.233, pp. 191-200 (in Russian, online copy: PDF). Podnieks K. M. [1976] The double-incompleteness theorem. Proceedings of Fourth All-Union Conference on Mathematical Logic, 1976, Kishinev, p.118 (in Russian, online copy: PDF, English translation: Section 6.2). Podnieks K. M. [1988a] Platonism, Intuition and the Nature of Mathematics. "Heyting'88. Summer School & Conference on Mathematical Logic. Chaika, Bulgaria, September 1988. Abstracts.", Sofia, Bulgarian Academy of Sciences, 1988, pp.50-51 (online copy: PDF). Podnieks K. M. [1988b] Platonism, Intuition and the Nature of Mathematics. Riga, Latvian State University, 1988, 23 pp. (in Russian). Podnieks K. M. [1981, 1992] Around Gödel's theorem. Latvian State University Press, Riga, 1981, 105 pp. (in Russian). 2nd edition: "Zinatne", Riga, 1992, 191 pp. (in Russian). Poincare H. [1908] Science et methode. Paris, 1908, 311 pp. (Russian translation available) Rashevsky P. K. [1973] (translated also as Rashevskii) On the dogma of natural numbers. Uspekhi matematicheskih nauk, 1973, vol.28, N 4, pp.243-246 (in Russian). See also Russian Mathematical Surveys, 1973, vol.28, N4, 143-148 (online copy). 6 1. Platonism, intuition and the nature of mathematics This chapter presents an extended the translation of the paper: K. Podnieks. Platonism, intuition and the nature of mathematics. In: Semiotika i informatika, Moscow, VINITI, 1990, Vol. 31, pp. 150-180 (in Russian). It was included also as the first chapter into my book "Around Gödel's theorem" published in 1992 in Russian. View onlinee