Math 570. Mathematical Logic

1. Syntax and semantics of propositional logic and first order logic.
2. Compactness theorem.
3. Systems of formal proofs and the completeness theorem.
4. 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.
5. 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.)

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Department of Mathematics University of Illinois at Urbana-Champaign 273 Altgeld Hall, MC-382 1409 W. Green Street, Urbana, IL 61801 USA Telephone: (217) 333-3350    Fax (217) 333-9576 office@math.uiuc.edu

Revised 6/12/99
Reviewed by Area Chair 6/20/05