Mathematical Logic, an Introduction by Peter Koepke Bonn, Winter 2019/20 Wann sollte die Mathematik je zu einem Anfang gelangen, wenn sie warten wollte, bis die Philosophie über unsere Grundbegrie zur Klarheit und Einmüthigkeit gekommen ist? Unsere einzige Rettung ist der formalistische Standpunkt, undenirte Begrie (wie Zahl, Punkt, Ding, Menge) an die Spitze zu stellen, um deren actuelle oder psychologische oder anschauliche Bedeutung wir uns nicht kümmern, und ebenso unbewiesene Sätze (Axiome), deren actuelle Richtigkeit uns nichts angeht. Aus diesen primitiven Begrien und Urtheilen gewinnen wir durch Denition und Deduction andere, und nur diese Ableitung ist unser Werk und Ziel. (Felix Hausdor, 12. Januar 1918) Table of contents 1 Introduction . 4 I First-order Logic and the Gödel Completeness Theorem . 4 2 The Syntax of rst-order logic: Symbols, words, and formulas . 4 2.1 Motivation: a mathematical statement . 5 2.2 Symbols . 5 2.3 Words . 6 2.4 Terms . 7 2.5 Formulas . 8 3 Semantics . 9 4 The satisfaction relation . 11 5 Logical implication and propositional connectives . 14 6 Substitution and term rules . 15 7 A sequent calculus . 20 8 Derivable sequent rules . 22 8.1 Auxiliary derived rules . 22 8.2 Introduction and elimination of ; ; ::: . 23 8.3 Formal proofs about . ._. .^. 24 9 Consistency . 25 1 2 Section 10 Term models and Henkin sets . 27 11 Constructing Henkin sets . 30 12 The completeness theorem . 34 13 The compactness theorem . 35 II Herbrand's Theorem and Automatic Theorem Proving . 38 14 Normal forms . 38 14.1 Negation normal form . 39 14.2 Conjunctive and disjunctive normal form . 39 14.3 Prenex normal form . 40 14.4 Skolem normal form . 41 15 Herbrand's theorem . 42 16 Computer implementation of symbolic logic . 44 16.1 Formulas . 44 16.2 Negation normal form . 45 16.3 CNF . 46 16.4 DNF . 49 16.5 Latin squares as a SAT problem . 50 17 Resolution . 51 18 Unikation . 53 III Set Theory . 57 19 Set theory . 57 19.1 The origin of set theory . 57 19.2 Set theoretic foundations of mathematics . 58 19.3 Class terms . 60 19.4 Properties of classes . 62 19.5 Set-theoretical axioms in class term notation . 63 20 Relations and functions . 63 20.1 Ordered pairs and cartesian products . 63 20.2 Relations . 64 20.3 Functions . 65 21 Ordinal numbers, induction and recursion . 66 21.1 -Induction . 67 2 21.2 Ordinal numbers . 67 21.3 Ordinal induction . 70 21.4 Ordinal recursion . 71 21.5 Ordinal arithmetic . 73 Table of contents 3 21.6 Sequences . 73 22 Cardinals . 74 23 Natural numbers and nite sets . 75 23.1 Natural numbers . 75 23.2 Finite cardinals . 77 23.3 Finite sets . 77 23.4 Finite sequences . 78 24 Innity . 78 24.1 ! . 79 24.2 Countable sets . 80 24.3 !-Sequences . 81 24.4 Uncountable sets . 82 25 Number systems . 82 25.1 Natural numbers . 86 25.2 Integer numbers . 87 25.3 Rational numbers . 88 25.4 Real numbers . 89 25.5 Discussion . 90 26 The Axiom of Choice . 91 27 The Alefs . 92 28 Cardinal Arithmetic . 93 29 Further Cardinal Arithmetic . 94 30 The Löwenheim-Skolem theorems . 96 IV Incompleteness . 99 31 Formalizing the metatheory in the object theory . 99 32 Sequences . 100 32.1 Symbols and Words . 100 32.2 Calculi . 100 32.3 Gödelization of nite sequences . 102 32.4 Finite and innite sets . 102 33 Formalizing syntax within ST . 102 34 ST in ST . 107 35 The undenability of truth . 107 36 Gödel's incompleteness theorems . 110 4 Section 2 37 The incompleteness of Zermelo-Fraenkel set theory . ..