# Math 770: Foundations of Mathematics (Instructor: Steffen Lempp)

This is an introductory graduate course in mathematical logic. We will discuss each of the four main subjects of logic: set theory, model theory, proof theory, and computability theory. The main topics will include the axioms of Zermelo-Fraenkel set theory with choice, ordinal and cardinal arithmetic, the syntax and semantics of first-order logic, a first-order proof calculus, the Completeness and Compactness Theorems, the Löwenheim-Skolem Theorems, and the Gödel Incompleteness Theorems.

### TEXTBOOK:

• Kenneth Kunen: The Foundations of Mathematics (College Publications)

### REFERENCE BOOKS:

• Ebbinghaus, Flum, Thomas: Mathematical Logic (Springer-Verlag)
• Hinman: Fundamentals of Mathematical Logic (A. K. Peters/CRC Press)
• Shoenfield: Mathematical Logic (Routledge)
• Enderton: A Mathematical Introduction to Logic (Academic Press, see class handouts on Presburger arithmetic)

### LECTURES:

Lectures will be given live at the regular time on Blackboard Collaborate Ultra, accessible from Canvas and posted there for later viewing.

### HOMEWORK:

Homework will be assigned intermittently throughout the semester and due at noon on the dte given below.
• HW # 1 DUE 9/18: Chapter I: 2.1, 6.13, 6.17, 7.16, 7.23 (in an online version of the book, these are 2.1, 6.12, 6.16, 7.16, 7.24)
• HW # 2 DUE 10/9: Chapter I: 8.10, 8.13, 9.6, 11.15, 11.33, 12.14, 12.15, 12.18 (in an online version of the book, these are 8.10, 8.13, 9.6, 11.13, 11.33, missing, 12.14, and missing)
• HW # 3 DUE 10/16: Chapter I: 13.20, 14.14, 14.24 (in an online version of the book, these are 13.20, 14.12, and missing)
• HW # 4 DUE 10/26: Chapter II: 4.7, 7.18, 7.19, 8.5, 8.20, 8.25
• HW # 5 DUE 11/11: Chapter II: 10.6, 11.10, 11.11, 11.13, 12.23
• HW # 6 DUE 11/30: Chapter VI: 3.4, 3.28, 3.29, 3.30

### TAKE-HOME FINAL EXAM:

Available on line here, due Friday, December 18, at noon.
Prepared by Steffen Lempp (@math.wisc.edu">lemppmath.wisc.edu)