The classical paradoxes, such as Russell's paradox and the Burali-Forti paradox in the foundations of mathematics, motivate an axiomatic description of set theory. Another motivation for the introduction of formal set theory is that the solution of several open problems in abstract analysis, measure theory and topology have been shown to depend on basic axioms for sets such as the continuum hypothesis, Martin's axiom or the Souslin hypothesis.
The course starts from naive set theory. In this context the theory of well-ordered sets, ordinals and cardinals is developed. After a brief revision of formal logic, the Zermelo Fraenkel (ZF) axioms are introduced. The axiom of choice and its most important implications in mathematics are discussed.
Consistency results are treated. The most famous example of a statement independent of ZFC is the continuum hypothesis but also other independent statements, coming from various branches of mathematics are discussed.
A foundation of category theory is presented: the existence of a universe is discussed and its relation with the axiom on the existence of a strongly inaccessible cardinal is treated.
1. Naive set theory
- Well ordering
- Paradoxes of Russell and Burali-Forti
2. Axiomatic set theory
- The language of set theory
- Zermelo Fraenkel axioms
3. The axiom of choice
- Equivalent formulations
- Applications in mathematics
- Equipotent sets
- Representation of infinite cardinals
- Continuum hypothesis
- Formal deduction
- Relative consistency
- Examples of independent statements
6. Well founded sets
- Axiom of foundation
- Constructible universe
7. Existence of a universe
-strongly inaccessible cardinals
8. Incompleteness theorem of Gödel