# Mathematics

## Logic: Exams

### Logic Comprehensive Exam

Before you decide to take the comprehensive exam in Mathematical Logic check out the overall structure of the comprehensive exam system.

#### Syllabus

The Comprehensive Exam in Mathematical Logic may contain problems in the following topics:

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, relations and functions representable in a theory, undecidability of various systems of arithmetic, undecidability of pure first order logic, and decidability of certain other theories.

#### Suggestions from a student who passed

• Make sure you know the topics above.
• If you have never taken a graduate course in logic or if several of the topics are not familiar to you, take Math 570 - this course is offered every Fall.
• Check out 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.).
• Solve as many problems from recently given comp exams as possible. Ask for help from logic graduate students if you need it.
• One more piece of advice (not only for comps in logic): GO AHEAD AND TAKE THE EXAMS! Don't wait until you are sure you know everything. WHY? Because:
• you never know what problems will be on a comp exam - maybe the comp exam you just didn't take was the one you would have passed.
• there is really no penalty for failing comps.
• you want to be done with comps as soon as possible.

### Logic Preliminary Exam

[The following policy statement was adopted by the faculty of the Logic Area in April, 2000, with revisions in September, 2002. It supplements the statement about the Preliminary Exams in the Guide for Graduate Students in Mathematics.]

The examining committee for a Preliminary Exam in Logic will consist of the thesis advisor and three other faculty members selected with the agreement of the thesis advisor. The committee should be officially appointed at least six weeks before the exam; this is done by the Graduate College on nomination by the Mathematics Department Director of Graduate Studies, who must approve the committee membership. A member of the committee other than the thesis advisor should be designated as Chair of the committee.

The exam lasts up to two hours including the student's presentation and is an oral exam. The exam should cover at least as much as the content of three advanced graduate semester courses (two in the expected area of thesis research and one for breadth). This means courses like Math 571, 573, 574, and 595, and other courses, including reading courses, at the same level. The focus of the exam will be the part of logic in which the thesis research is going to be carried out, but breadth is also required.

The exam should normally cover two of the four principal subdivisions of logic (model theory, set theory, computability theory, and proof theory). A related topic at a similar level may be substituted for one of these areas. The exam will include questions in depth about the proposed thesis area and basic questions about a second advanced topic in logic. Alternate topics meeting the same general standards as what is described here are possible, with the approval of the exam committee. [This assumes the student has passed the Comprehensive Exam in Logic (based on Math 570). Otherwise, the Preliminary Exam will also include questions about the content of Math 570, in addition.]

In addition to the oral exam, the Preliminary Exam in Logic should include a presentation by the student of some material from the proposed area of thesis research. This can either be done in one of our seminars or at the beginning of the oral exam (in which case the lecture should not last more than about 30 minutes). During this presentation the student should demonstrate mastery of detail as well as understanding of the overall structure and aims of the material being presented. In either case, the actual question period during the Preliminary Exam will not last more than one hour and a half.

The content of the Preliminary Exam including the nature of the student's presentation should be agreed to by the examination committee at least one month before the exam.