Math 414: MATHEMATICAL LOGIC, Section C13/C14

Instructor: L. van den Dries

Office hours: Monday and Friday, 8:30AM to 9:30AM (Altgeld Hall, 308).

No previous study of logic is assumed, but we begin with a brief introduction of elementary set theory as an efficient language to communicate mathematics, including the mathematical logic of this course. After that the main topics are:

- Propositional Logic, its syntax and semantics, and natural deduction as a proof system. The main result is a a completeness theorem for propositional logic. It says that semantic validity (using truth tables) is equivalent to provability in the proof system of natural deduction. Propositional Logic is very limited, but it gives a good preparation for what comes next, namely Predicate Logic.
- The same things for Predicate Logic. Predicate Logic has a claim on being a complete system of logic as used in mathematical reasoning. Here the semantics include the notion of a structure being a model of a set of sentences. The main result is Godel's completeness theorem, again for a proof system based on natural deduction. We include applications to specific mathematical theories.

There will be three midsemester exams, a final exam, and regular homework, the latter due each Monday at the beginning of class. These count to your course grade as follows: HW: 10/100, midterms: 30/100, final: 60/100.

Prerequisites: Math 347, or Math 348.

Text: Dirk van Dalen, Logic and Structure, fifth edition, Universitext, Springer

Some historical remarks about mathematical logic, and an introduction to naive set
theory as covered the beginning of the course are contained in the first 12 pages
of my Logic Notes (Math 570), which can be found under "Lecture Notes" at
https://faculty.math.illinois.edu/~vddries/

First midterm: Monday February 18, 7PM-8.30PM, Lincoln Hall, room 1090. This covers: The notes on Sets and Maps from the first week, and Chapters 1 and 2 of the book by van Dalen, except for section 2.5. Also any of the files posted like CNF and solutions to homeworks.

Second midterm: Monday April 1, 7PM-8.30PM, English Building, room 108. This covers: Section 2.5, and Chapter 3. Also any of the solutions to homeworks. You can bring one sheet with both sides covered by your own summary of the material. HW9 will be assigned on Monday March 25, and will be due on the Friday of that week:March 29. Below I list a file with further info about the contents of the midterm

Final Exam: Friday May 3, 1:30 to 4:30, 156 Henry Administration Building (next door from Altgeld Hall). This will cover the notes on set theory distributed at the beginning of the course plus some further notes on cardinal numbers distributed Wednesday April 24, plus the material from the book up to and including Section 4.2. I will review this material on Monday, May 1, during class. You can bring one sheet with facts about the material to the exam (using both sides).Last modified April 19