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

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Unformatted text preview: Topics in Logic and Foundations: Spring 2005 Stephen G. Simpson Copyright c circlecopyrt 2005 First Draft: April 29, 2005 This Draft: November 1, 2005 The latest version is available at http://www.math.psu.edu/simpson/notes/. Please send corrections to <simpson@math.psu.edu> . 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 2005. 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. These notes were typeset by the students in the course: John Ethier, Esteban Gomez-Riviere, David King, Carl Mummert, Michael Rowell, Chenying Wang. In addition, the notes were revised and polished by Stephen Simpson. Contents Contents 1 1 Unsolvability of Hilberts Tenth Problem 3 1.1 Hilberts Tenth Problem . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 1 Relations and Functions . . . . . . . . . . . . . . . . . . . . . 5 1.3 Diophantine Relations and Functions . . . . . . . . . . . . . . . . 7 1.4 Bounded Universal Quantification . . . . . . . . . . . . . . . . . 9 1.5 The Pell Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.2 Divisibility Properties of y n . . . . . . . . . . . . . . . . . 14 1.5.3 Congruence Properties of x n . . . . . . . . . . . . . . . . 15 1.5.4 Diophantine Definability of x n and y n . . . . . . . . . . . 16 1.6 Proof of the Main Lemma . . . . . . . . . . . . . . . . . . . . . . 17 2 Unsolvability of the Word Problem for Groups 21 2.1 Finitely Presented Semigroups . . . . . . . . . . . . . . . . . . . 21 2.2 The Boone Group . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 HNN Extensions and Brittons Lemma . . . . . . . . . . . . . . . 30 2.4 Free Products With Amalgamation . . . . . . . . . . . . . . . . . 32 2.5 Proof of 3 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6 Proof of Brittons Lemma . . . . . . . . . . . . . . . . . . . . . . 36 2.7 Proof of 2 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.8 Some Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.9 Unsolvability of the Triviality Problem . . . . . . . . . . . . . . . 41 3 Recursively Enumerable Sets and Degrees 44 3.1 The Lattice of R.E. Sets . . . . . . . . . . . . . . . . . . . . . . . 44 3.2 Many-One Completeness . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Creative Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4 Simple Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5 Lattice-Theoretic Properties . . . . . . . . . . . . . . . . . . . . . 54 3.6 The Friedberg Splitting Theorem . . . . . . . . . . . . . . . . . . 55 3.7 Maximal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3....
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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 Foundations - Topics in Logic and...

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