The theory of the foundations of mathematics - 1870 to 1940 - Mark Scheffer (Version 1.0) 2 3 . Mark Scheffer, id. 415968, e-mail:
[email protected]. Last changes: March 22, 2002. This report is part of a practical component of the Com- puting Science study at the Eindhoven University of Technology. 4 To work on the foundations of mathematics, two things are needed: Love and Blood. - Anonymous quote, 2001. Contents 1 Introduction 9 2 Cantor’s paradise 13 2.1Thebeginningofset-theory................... 13 2.2Basicconcepts........................... 15 3 Mathematical constructs in set-theory 21 3.1Somemathematicalconcepts................... 21 3.2Relations.............................. 23 3.3Functions............................. 29 3.4 Induction Methods . 32 3.4.1 Induction . 32 3.4.2 Deduction......................... 33 3.4.3 The principle of induction . 34 3.5Realnumbers........................... 45 3.5.1 Dedekind’scuts...................... 46 3.5.2 Cantor’schainsofsegments............... 47 3.5.3 Cauchy-sequences..................... 48 3.5.4 Propertiesofthethreedefinitions............ 50 3.6Infinitesets............................ 51 3.7TheContinuumHypothesis................... 60 3.8 Cardinal and Ordinal numbers and Paradoxes . 63 3.8.1 Cardinal numbers and Cantor’s Paradox . 63 3.8.2 Ordinal numbers and Burali-Forti’s Paradox . 65 4 Peano and Frege 71 4.1Peano’sarithmetic........................ 71 4.2Frege’swork............................ 74 5 6 CONTENTS 5 Russell 79 5.1 Russell’s paradox . 82 5.2Consequencesandphilosophies................. 88 5.3ZermeloFraenkel......................... 92 5.3.1 Axiomatic set theory . 92 5.3.2 Zermelo Fraenkel (ZF) Axioms . 93 6 Hilbert 99 6.1Hilbert’sprooftheory.......................101 6.2Hilbert’s23problems.......................110 7 Types 113 7.1 Russell and Whitehead’s Principia Mathematica . 113 7.2Ramsey,HilbertandAckermann.................119 7.3Quine...............................121 8G¨odel 123 8.1 Informally: G¨odel’s incompleteness theorems .