Foundations of Mathematics - Copyright c circlecopyrt...

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Unformatted text preview: Copyright c circlecopyrt 1995–2007 by Stephen G. Simpson Foundations of Mathematics Stephen G. Simpson February 10, 2008 Department of Mathematics The Pennsylvania State University University Park, State College PA 16802 [email protected] This is a set of lecture notes for my course, Foundations of Mathematics I, offered as Mathematics 558 at the Pennsylvania State University, most recently in Spring 2007. 1 Contents 1 Computable Functions 6 1.1 Primitive Recursive Functions . . . . . . . . . . . . . . . . . . . . 6 1.2 The Ackermann Function . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Computable Functions . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Partial Recursive Functions . . . . . . . . . . . . . . . . . . . . . 23 1.5 The Enumeration Theorem . . . . . . . . . . . . . . . . . . . . . 25 1.6 Consequences of the Enumeration Theorem . . . . . . . . . . . . 30 1.7 Unsolvable Problems . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.8 The Recursion Theorem . . . . . . . . . . . . . . . . . . . . . . . 39 1.9 The Arithmetical Hierarchy . . . . . . . . . . . . . . . . . . . . . 41 2 Undecidability of Arithmetic 48 2.1 Terms, Formulas, and Sentences . . . . . . . . . . . . . . . . . . . 48 2.2 Arithmetical Definability . . . . . . . . . . . . . . . . . . . . . . . 50 2.3 G¨ odel Numbers of Formulas . . . . . . . . . . . . . . . . . . . . . 57 3 The Real Number System 60 3.1 Quantifier Elimination . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2 Decidability of the Real Number System . . . . . . . . . . . . . . 66 4 Informal Set Theory 69 4.1 Operations on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Cardinal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3 Well-Orderings and Ordinal Numbers . . . . . . . . . . . . . . . 74 4.4 Transfinite Recursion . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.5 Cardinal Numbers, Continued . . . . . . . . . . . . . . . . . . . . 81 4.6 Cardinal Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.7 Some Classes of Cardinals . . . . . . . . . . . . . . . . . . . . . . 85 4.8 Pure Well-Founded Sets . . . . . . . . . . . . . . . . . . . . . . . 87 4.9 Set-Theoretic Foundations . . . . . . . . . . . . . . . . . . . . . . 88 5 Axiomatic Set Theory 92 5.1 The Axioms of Set Theory . . . . . . . . . . . . . . . . . . . . . . 92 5.2 Models of Set Theory . . . . . . . . . . . . . . . . . . . . . . . . 96 2 5.3 Transitive Models and Inaccessible Cardinals . . . . . . . . . . . 99 5.4 Constructible Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.6 Independence of CH . . . . . . . . . . . . . . . . . . . . . . . . . 111 6 Topics in Set Theory 114 6.1 Stationary Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.2 Large Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.3 Ultrafilters and Ultraproducts . . . . . . . . . . . . . . . . . . . . 116Ultrafilters and Ultraproducts ....
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This note was uploaded on 03/30/2008 for the course MATH 563 taught by Professor Simpson,stephen during the Fall '05 term at Pennsylvania State University, University Park.

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Foundations of Mathematics - Copyright c circlecopyrt...

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