## Spring 2019

Math 414: MATHEMATICAL LOGIC, Section C13/C14

## MWF from 10:00AM to 10:50AM in Room 241 Altgeld Hall

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/

Next I list various links to the syllabus, homework assignments (numbered as HW1, HW2, and so on), solutions of these assignments (HW1s, HW2s, and so on),
exam dates, and the like.
syllabus

HW1

HW1solutions

HW2

HW2solutions

HW3

HW3solutions

CNF

HW4

HW4solutions

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.

Last modified February 8, 2019