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 . 123 8.2 Formally: G¨odel’sIncompletenessTheorems..........127 8.2.1 On formally undecidable propositions . 127 8.2.2 The impossibility of an ‘internal’ proof of consistency . 130 8.2.3 G¨odel numbering and a concrete proof of G1, G2 and G3 131 8.3 G¨odel’s theorem and Peano Arithmetic . 132 8.4Consequences...........................134 8.5 Neumann-Bernays-G¨odelaxioms.................135 9 Church and Turing 141 9.1TuringandTuringMachine...................141 9.2ChurchandtheLambdaCalculus................153 9.3TheChurch-Turingthesis....................166 10 Conclusion 169 A Timeline and Images 181 CONTENTS 7 Mathematical Notations Many different notations have been developed for set theory and logic. Most notations that we have used are standard today; other notations that we have used are introduced in the text. Mathematical Logic symbol meaning also described as ∧ conjuction and ∨ disjunction (inclusive) or ¬ negation not ϕ(x) propositional function → implication if ... then ↔ bi-implication if and only if, iff ≡ equivalence is equivalent to ∀ universal quantifier for all ∃ existential quantifier exists ∃! one-element existential quantifier exists a unique In most places we have chosen to use the following notation1 to denote quantifications: (relation : range : term) denotes the relationship over a set of termsrangingoverrange Consider a general pattern (Qx : ϕ(x0,...,xn):t(x0,...,xn)), with Q aquantifier,ϕ a boolean expression in terms of the dummies x0, ..., xn, and t(x0,...,xn) the term of the quantification. The quantification is the accumulation of values t(x0,...,xn) using an operator or relation indicated by Q, over all values (x0,...,xn) for which ϕ(x0,...,xn)holds. 1Notation originally due to E.W. Dijkstra. 8 CONTENTS This notation is suitable for formal manipulation and unambiguous in the sense that it explicitly indicates the quantifier Q, the dummies and the range of the dummies that is indicated by the boolean expression ϕ (i.e. it exactly determines the domain of the quantification). This allows us to reason about general properties of quantifications, in a way in which the (scopes of the) bound variables are clearly identified. Note that this type of quantification is only suitable for binary operations that are symmetric and associative. Example: ( x :0≤ x ≤ 5:x2) = 02 +12 +22 +32 +42 +52 = 5 x2 x=0 Example: (∃x : x ∈ N : x3 − x2 =18) ≡ ‘there exists a natural number x such that x3 − x2 =18’ If the term ranges over all possible values of the variable (here : x), or if it is clear what the range of a variable is, we can omit it. Example: (∀x : true : x ∈ A → x ∈ B) ≡ (∀x :: x ∈ A → x ∈ B) ≡ ‘all elements of A are also elements of B’ Chapter 1 Introduction Pure mathematics is, in its way, the poetry of logical ideas. - Albert Einstein This report covers the most important developments and theory of the foundations of mathematics in the period of 1870 to 1940. The tale of the foundations is fairly familiar in general terms and for its philosophical con- tent; here the main emphasis is laid on the mathematical theory. The history of the foundations of mathematics is complicated and is a many-sided story; with this article I do not aim to give a definitive or complete version, but to capture what I consider the essence of the theoretical developments, and to present them in a clear and modern setting. Some basic mathematical knowledge on set-theory and logics are presupposed. By the middle of the nineteenth century, certain logical problems (for example paradoxes around the notions of infinity, the infinitesimal and con- tinuity) at the heart of mathematics had inspired a movement, led by German mathematicians, to provide mathematics with more rigorous foundations. This is where the theory of this report begins, with the emergence of set theory by the German mathematician Cantor. In section 2.1 we informally describe how work on a problem concerning trigonometric series gradually led Cantor to his theory of sets (section 2.2). As a result of the work of Weierstrass, Dedekind and Cantor, pure mathematics had been provided with much more sophisticated foundations. The notion of infinitesimal had been banished, ‘real’ numbers had been provided with a logically consistent 9 10 CHAPTER 1. INTRODUCTION definition (section 3.5), continuity had been redefined and, more controver- sially, a whole new branch of arithmetic had been invented which addressed itself to the problems (e.g. paradoxes) of infinity (sections 3.6, 3.7). In 1895 Cantor discovered a paradox (section 3.8.1) that he did not publish but communicated to Hilbert in 1896. In 1897 it was rediscovered in a slightly different form by Burali-Forti (section 3.8.2). Cantor and Burali-Forti could not resolve this paradox, but it was not taken so seriously, partly because the paradoxes appeared in a rather technical region. The Italian mathematician Peano (section 4.1) was able to show that the whole of arithmetic could be founded upon a system that uses three basic notions and five initial axioms. At the same time the German mathematician Frege (section 4.2) worked on developing a logical basis for mathematics. Just as Peano, Frege wanted to put mathematics on firm grounds. But Frege’s grounds were strictly logic; he followed a development later called logicism, also known as the development of so-called mathematical logic. The British mathematician Russell noted Peano’s work and later that of Frege. Soon thereafter he showed (section 5.1) how finite descriptions like ‘set of all sets’ could be self-contradictory (i.e. paradoxical) and pointed out the difficulties that arose with self-referential terms. This paradox that Russell found existed not only in specific technical regions but in all of the axiomatic systems underlying mathematics at the same time (section 5.1). But since the paradoxes could be avoided in most practical applications of set theory, the belief in set theory as a proper foundation of mathematics remained. Axiomatic set theory (section 5.3.1) was an attempt to come to a theory without paradoxes. Various responses to the paradox (section 5.2) led to new sets of axioms for set theory. The two main approaches are by the German mathematicians Zermelo and Fraenkel (section 5.3), and by the Hun- garian von Neumann, the Hungarian-Austrian G¨odel and the Briton Bernays (section 8.5). It also led to the emergence of the ‘intuitionistic’ philosophy of mathematics by the Dutch mathematician Brouwer (not covered here) and to a theory of types, proposed by Russell himself with the help of his for- mer teacher, the English mathematician Whitehead. Despite of the paradox Russell and Whitehead still claimed that all mathematics could be founded on a mathematical logic; this believe was given a definite presentation in their work ‘Principia Mathematica’ (section 7.1). Various consequences fol- lowed (section 7.3) and new conceptions of logic arose (by Wittgenstein and 11 Ramsey, see section 7.2). At the turn of the century, the German mathematician David Hilbert listed certain important problems concerning the foundations of mathema- tics and mathematics in general (section 6.2. To overcome paradoxes and other problems that arose in existing systems, Hilbert developed a theory of axiomatic systems (section 6.1). He then stimulated his student Zermelo in using this axiomatic method to develop as first a set of axioms for set theory (section 5.3.2). Hilbert had since then made more precise demands on any proposed set of axioms for mathematics (section 6.1) in terms of consistency, completeness and decidability. In 1931 G¨odel had shown that consistency and completeness could not both be attained (chapter 8). G¨odel’s work left outstanding Hilbert’s ques- tion of decidability. The English mathematician Turing proved in 1936 that there are undecidable problems, by giving the so-called halting problem that cannot be solved by any algorithm (section 9.1), after formalizing the no- tion of algorithm with his concept of the Turing Machine. The American mathematician Church (independently) obtained the same result but with another formalization of the notion of an algorithm, using his computational model of lambda calculus (section 9.2).