Basics of Mathematical Logic

Last updated: 2024-01-02

Course Info

Abstract

In this course, we explore the significant achievements and developements in the field of mathematical logic from the last century. Topics include first-order logic, recursion theory and computability, Gödel’s incompleteness theorems, model theory, and more.

Time

Every Monday, 9:50 am – 12:15 pm.

Classroom

A3-2a 302 (Zoom: 537 192 5549, Pw: BIMSA)

References

Robert S. Wolf, A Tour Through Mathematical Logics
Yu. I. Manin & B. Zilber, A Course in Mathematical Logic for Mathematicians
Stanford Encyclopedia of Philosophy (SEP)

Schedule and Notes

Date	Topic	Reading	Notes
Sep. 18	Introduction, propositional logic	Ref. 1, 1.1-1.3	-
Sep. 25	First-order languages and formal proof	Ref. 1, 1.4
Oct. 02	Holiday	-	-
Oct. 09	First-order theories, prenex form and syntactic complexity	Ref. 1, 1.5-1.6
Oct. 16	Axiomatic Set theory	Ref. 1, 2.1-2.3
Oct. 23	Ordinals and cardinals	Ref. 1, 2.4-2.5
Oct. 30	Recursion theory and computability	Ref. 1, 3.1-3.3
Nov. 06	Complexity theory	Ref. 1, 3.4-3.5
Nov. 13	Godel’s incompleteness theorems	Ref. 1, 4.1-4.2
Nov. 20	Incompleteness theorems	Ref. 1, 4.3-4.4
Nov. 27	Model theory	Ref. 1, 5.1-5.3
Dec. 04	Preservation theorems	Ref. 1, 5.4
Dec. 11	Saturation and completeness	Ref. 1, 5.5
Dec. 18	Category theory intro
Dec. 25	Hilbert spaces and monoidal category
Jan. 01	Holiday	-	-
Jan. 08	Monoidal category, cont'
~~Jan. 15~~ Jan. 08	Linear structure

Priliminary syllabus

First order logic
- Propositional logic
- Quantifiers
- First-order languages and theories
- Normal forms and complexity
Recursion theory and computability
- Primitive recursive functions
- Turing machines and recursive functions
- Undecidability
- Complexity theory
Godel’s incompleteness theorems
- The arithmetization of formal theories
- Incompleteness theorems
Model theory
- Main theorems: Godel’s completeness theorem, compactness theorem, Lowenheim-Skolem-Tarski theorem
- Preservation theorems
- Complete theoreis