Topics in Logic and - Topics in Logic and Foundations Spring 2004 Stephen G Simpson Copyright c circlecopyrt 2004 First Draft This Draft November 1

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Unformatted text preview: 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. 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 . . . . . . . . . . . . . . . . .Post’s Theorem and its corollaries ....
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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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Topics in Logic and - Topics in Logic and Foundations Spring 2004 Stephen G Simpson Copyright c circlecopyrt 2004 First Draft This Draft November 1

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