{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Foundations of Mathematics

Foundations of Mathematics - Copyright c 19952007 by...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 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
Background image of page 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 . . . . . . . . . . . . . . . . . . . . 116 6.4 Measurable Cardinals . . . . . . . . . . . . . . . . . . . . . . . . 119 6.5 Ramsey’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
List of Figures 1.1 Register Machine Instructions . . . . . . . . . . . . . . . . . . . . 18 1.2 An Addition Program . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3 The Initial Functions . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.4 Generalized Composition . . . . . . . . . . . . . . . . . . . . . . 20 1.5 A Multiplication Program . . . . . . . . . . . . . . . . . . . . . . 21 1.6 Primitive Recursion . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.7 Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.8 A Program with Labeled Instructions . . . . . . . . . . . . . . . 27 1.9 Incrementing P i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.10 Decrementing P i . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.11 Stopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.12 Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4
Background image of page 4
List of Tables 1.1 The Ackermann branches . . . . . . . . . . . . . . . . . . . . . . . 14 5
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Chapter 1 Computable Functions We use N to denote the set of natural numbers, N = { 0 , 1 , 2 ,... } . For k 1, the k -fold Cartesian product N × ... × N bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright k is denoted N k . A k -place function is a function f : N k N and is sometimes indicated with the lambda-notation , f = λx 1 · · · x k [ f ( x 1 ,...,x k )] . A number-theoretic function is a k -place function for some k 1. The purpose of this chapter is to define and study an important class of number-theoretic functions, the recursive functions (sometimes called the com- putable functions). We begin with a certain subclass known as the primitive recursive functions. 1.1 Primitive Recursive Functions Loosely speaking, a recursion is any kind of inductive definition, and a primitive recursion is an especially straightforward kind of recursion, in which the value of a number-theoretic function at argument x + 1 is defined in terms of the value at argument x . For example, the factorial function λx [ x ! ] is defined by the primitive recursion equations 0! = 1, ( x + 1)! = x !( x + 1). A number- theoretic function is said to be
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}