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Topics in Logic and Foundations

Topics in Logic and Foundations - Topics in Logic and...

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Topics in Logic and Foundations: Spring 2004 Stephen G. Simpson Copyright c circlecopyrt 2004 First Draft: April 30, 2004 This Draft: November 1, 2005 The latest version is available at http://www.math.psu.edu/simpson/notes/. Please send corrections to <[email protected]> . This is a set of lecture notes from a 15-week graduate course at the Penn- sylvania State University taught as Math 574 by Stephen G. Simpson in Spring 2004. The course was intended for students already familiar with the basics of mathematical logic. The course covered some topics which are important in contemporary mathematical logic and foundations but usually omitted from introductory courses at Penn State. These notes were typeset by the students in the course: Robert Dohner, Esteban Gomez-Riviere, Christopher Griffin, David King, Carl Mummert, Heiko Todt. In addition, the notes were revised and polished by Stephen Simpson.
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Contents Contents 1 1 Computability in core mathematics 4 1.1 Review of computable functions . . . . . . . . . . . . . . . . . . . 4 1.1.1 Register machines . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Recursive and partial recursive functions . . . . . . . . . . 5 1.1.3 The μ -operator . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Introduction to computable algebra . . . . . . . . . . . . . . . . . 7 1.2.1 Computable groups . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 Computable fields . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Finitely presented groups and semigroups . . . . . . . . . . . . . 10 1.3.1 Free groups . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.2 Group presentations and word problems . . . . . . . . . . 11 1.3.3 Finitely presented semigroups . . . . . . . . . . . . . . . . 13 1.3.4 Unsolvability of the word problem for semigroups . . . . . 14 1.4 More on computable algebra . . . . . . . . . . . . . . . . . . . . 15 1.4.1 Splitting algorithms . . . . . . . . . . . . . . . . . . . . . 15 1.4.2 Computable vector spaces . . . . . . . . . . . . . . . . . . 16 1.5 Computable analysis and geometry . . . . . . . . . . . . . . . . . 16 1.5.1 Computable real numbers . . . . . . . . . . . . . . . . . . 16 1.5.2 Computable sequences of real numbers . . . . . . . . . . . 17 1.5.3 Effective Polish spaces . . . . . . . . . . . . . . . . . . . . 18 1.5.4 Examples of effective Polish spaces . . . . . . . . . . . . . 19 1.5.5 Effective topology and effective continuity . . . . . . . . . 20 2 Degrees of unsolvability 22 2.1 Oracle computations . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Relativization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 The arithmetical hierarchy . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Turing degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5 The jump operator . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.6 Finite approximations . . . . . . . . . . . . . . . . . . . . . . . . 29 2.7 Post’s Theorem and its corollaries . . . . . . . . . . . . . . . . . 32 2.8 A minimal Turing degree . . . . . . . . . . . . . . . . . . . . . . 34 1
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2.9 Sacks forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.10 Homogeneity of Sacks forcing . . . . . . . . . . . . . . . . . . . . 42 2.11 Cohen genericity . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3 Models of set theory 44 3.1 Countable transitive models . . . . . . . . . . . . . . . . . . . . . 44 3.2 Models constructed by forcing . . . . . . . . . . . . . . . . . . . . 44 3.3 An example: Cohen forcing . . . . . . . . . . . . . . . . . . . . . 46 3.4 Properties of generic extensions . . . . . . . . . . . . . . . . . . . 47 3.5 Blowing up the continuum . . . . . . . . . . . . . . . . . . . . . . 48 3.6 Preservation of cardinals and the c.c.c . . . . . . . . . . . . . . . . 49 3.7 Forcing the Continuum Hypothesis . . . . . . . . . . . . . . . . . 51 3.8 Additional models obtained by forcing . . . . . . . . . . . . . . . 52 4 Absoluteness and constructibility 54 4.1 Absoluteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Trees and well foundedness . . . . . . . . . . . . . . . . . . . . . 55 4.3 The Shoenfield Absoluteness Theorem . . . . . . . . . . . . . . . 58 4.4 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.5 Constructible sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.6 Constructible reals . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.7 Relative constructibility . . . . . . . . . . . . . . . . . . . . . . . 63 5 Measurable cardinals 65 5.1 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2 The closed unbounded filter . . . . . . . . . . . . . . . . . . . . . 66 5.3 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.4 Ultraproducts and ultrapowers . . . . . . . . . . . . . . . . . . . 68 5.5 An elementary embedding of V . . . . . . . . . . . . . . . . . . . 70 5.6 Largeness and normality . . . . . . . . . . . . . . . . . . . . . . . 72 5.7 Ramsey’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.8 Indiscernibles and EM-sets . . . . . . . . . . . . . . . . . . . . . 75 5.9 Measurable cardinals and L . . . . . . . . . . . . . . . . . . . . . 78 5.10 The # operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6 Determinacy 84 6.1 Games and determinacy . . . . . . . . . . . . . . . . . . . . . . . 84 6.2 Open and Borel determinacy . . . . . . . . . . . . . . . . . . . . 86 6.3 Projective sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.4 Consequences of projective determinacy . . . . . . . . . . . . . . 89 6.5 Turing degree determinacy . . . . . . . . . . . . . . . . . . . . . . 93 6.6 Σ 1 1 determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.7 Hyperarithmetic theory . . . . . . . . . . . . . . . . . . . . . . .
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