1. komponenta

Lecture type	Total
Lectures	30
Exercises	30

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES: The course is divided into two parts. Naive set theory is studied in the first part. The notions of countable and uncountable sets are considered and the notions related to ordered sets are defined. The main aim of the first part is to motivate introduction of axioms. In the second part is considered Zermelo-Fraenkel set theory. The notions of ordinal and cardinal numbers are defined, and the enumeration theorem is proved. At the end the axiom of choice is studied.

COURSE DESCRIPTION AND SYLLABUS:
Naive set theory
1. Introduction. History of set theory. Axiom of extensionality. Russell's paradox. Axiom of choice. Idea of cumulative hierarchy. Axiom of empty set and pairing axiom. Ordered pair. Definitions of relation and function. Union axiom and powerset axiom.
2. Equipotent sets. Finite and infinite sets. Countable sets.
3. Countable union of countable sets is countable set (application of axiom of choice). Set of rational numbers is countable. Every infinite set consists a countable subset (application of axiom of choice).
4. Uncountable sets. Set of reals is uncountable. Cardinal numbers (naive approach). Basic Cantor theorem. Arithmetic of cardinal numbers.
5. Knaster - Tarski fixed point theorem. Banach's lemma. Cantor - Bernstein - Schroder theorem.
6. Binary relations: reflexive, ireflexive, symmetric, antisymmetric and transitive. Equivalence relations. Partial order sets. Comparable elements. Chain, maximum and minimum, the greatest and the least elements, upper bound, supremum and infimum. Linear order sets. Order-preserving functions. Isomorphism.
7. Dense partial order sets. Cuts. Dedekind style continuity. Theorems on ordered characteristics of sets Q i R.
8. Wellorder sets. Basics properties of wellorder sets. Principle of transfinite induction. Axiom of foundation.
Axiomatic set theory
1. Set of natural numbers: inductive set, natural numbers, axiom of infinity, axiom schema of separation. Axiom of induction. Transitive sets. Basics properties. Every natural number is transitive set. Dedekind recursion theorem.
2. The sets of numbers: integers, rational and real numbers. Von Neumann definition of ordinals.
3. Basics properties of ordinal numbers. The enumeration theorem.
4. The ordering of ordinal numbers. Successor and limit ordinal. Recursion theorem. The addition and multiplication of ordinal numbers. Theorem on subtraction.
5. Ordinal exponentiation. Normal form theorem for ordinal numbers. Cantor normal form. Goodstein theorem.
6. Cardinal numbers. Arithmetic of cardinal numbers. Konig's lemma. Alephs. Axiom schema of replacement.
7. Axiom of choice. Cartesian product. Russell's axiom. Zorn's lemma. Hausdorff's maximal principle. Teichmuller - Tukey lemma. Zermelo's theorem. Hartog's theorem. Proofs of simple equivalences. Banach - Tarski paradox.

Enrollment :
Passed : Elementary mathematics 1
Passed : Linear algebra 2
Passed : Mathematical analysis 2

Mandatory course - Regular study - Mathematics