# 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)