# Set Theory

Jan Hubička, hubicka@kam.mff.cuni.cz

Consultations by email arrangement. Use "Set Theory" in subjects of emails.

• Lecture: Tue 14:00, S4
• Tutorials: Andres Aranda, web
• Czech Class: Honza Kyncl, web

## Topics covered

Feb 14
• Brief history of proofs in mathematics (Pythagoras theorem and its visual proof, Euclid's elements and a gap in proof of Proposition I). Not required for exam.
• Why do we want to study set theory? We want solid foundation to mathematics to resolve some problems:
• Russel's paradox: set X is extraordinary if X ∈ X, ordinary otherwise. Let O be the set of all ordinary sets. Is O ∈ P?
• Russel's paradox in popular language: "There is one barber in a town. He shaves exactly those men, who do not shave themselves. Does he shave himself?"
• Berry paradox: "X is the smallest positive integer not definable in under sixty letters." Arguably X both exists and does not exist.
• There are also highly counter-intuitive but correct results such at the Banach-Tarski paradox on sphere doubling.
In 1900 these problems culminated thanks to the Hilbert's programme and led to the foundational crisis.
• Brief history: Bolzano introduced set theory to study infinity. Cantor's work initiated modern set theory. Russel-Whitehead: Principia mathematica and its proof that 1+1=2 as an attempt to resolve the foundational crisis, Goedel and his incompleteness results (very briefly)
• Brief crash-course on first order formulas. We will be most of time informal and use English, however we keep in mind that the underlying language is the predicate logic and everything we do can be expressed by it.
• Axiomatic approach: we will follow Zermelo--Fraenkel axiomatization from 1908 (Van Neumann-Gödel-Bernays is other popular)
• The Axiom of Existence (Axiom existence): empty set exists. We found way to express it as formula
• The Axiom of Extensionality (Axiom extenzionality): If two sets have precisely same elements they are equal. Again we wrote a formula
• Thorem: Empty set is unique. Proof.

## Literature

Many recommended books are mentioned in the SIS description of the class. We loosely follow the book by Hrbacek and Jech. You may want to consult notes of Václav Končický (in Czech).