## Math 570. Mathematical Logic Instructor Syllabus

- Syntax and semantics of propositional logic and first order logic.
- Compactness theorem.
- Systems of formal proofs and the completeness theorem.
- Basic elements of model theory (completeness of theories, categoricity, quantifier elimination) and examples such as dense linear orderings, vector spaces, algebraically closed fields, and simple fragments of arithmetic.
- Incompleteness theorem and related topics, including: basic properties of computable functions, undecidability of various systems of arithmetic, undecidability of pure first order logic, and decidability of certain other theories.

NOTES:

(1) Students should have a detailed understanding of the proofs of the theorems covered by this syllabus, especially the Goedel completeness and incompleteness theorems.

(2) Students should consult past exams for sample questions; however, note that logic comprehensive exams given prior to June, 1998, covered both Math 410 and another course, and were based on a slightly different syllabus for Math 410. Syllabus topics not covered fully in a given offering of Math 410 may nonetheless be included on the comprehensive exam. If in doubt or needing help for self-study, please consult one of the logic faculty.

**REFERENCES: **

Chapters 1-3 (except sections 1.6, 2.7, 2.8, 3.6, and 3.7) of H. Enderton, A Mathematical Introduction to Logic, or the corresponding material in J. Shoenfield, Mathematical Logic. (A treatment of algebraically closed fields is in Shoenfield but not Enderton.)