Instructor: Dr. Lajos SOUKUP
Homepage of the course: http://www.renyi.hu/~soukup/set_14s.html
Prerequisite: Some familiarity with "higher" mathematics. No specific knowledge is expected.
Course description
- We get an insight how set theory can serve as the foundation of mathematics: all mathematical concepts, methods, and results can be represented within set theory.
- We study how to build up a rich mathematical theory from simple axioms.
- We learn how to use set theory as a powerful tool in algebra, analysis, and even geometry
- Since set theory is also an independent branch of mathematics, like algebra or geometry, with its own subject matter, basic results, open problems, the course tries to catch a glimpse of some results and problems from contemporary set theory, especially from infinite combinatorics.
Topics:
- Naive set theory. Basic principles:
- Extensionality: Two sets are equal if and only if they have the same elements.
- General principle of comprehension: If P(x) is a property, then there is a set Y={x:P(x)} of all elements having property P.
- Countable and uncountable sets. An application: there are uncountably many transcendental real numbers.
- Inductive constructions. The infinite Ramsey Theorem.
- The fall of naive set theory:
The general principle of comprehension leads to contradiction.
- Russel's Paradox: Does the set of all those sets that do not contain themselves contain itself?
- The solution: keep Extensionality, and replace the faulty General Comprehension with some weaker hypotheses, axioms, which are necessary for the proofs of the fundamental
results and seemingly free of contradiction.
For example,
- Axiom of Pairing: For any set A and set B, there is a set C such that x ∈ C if and only if x=A or x=B.
- Axiom Schema of Separation: Let P(x) be a property of x. For each set A there is set B such that x ∈ B if and only if x ∈ A and P(x).
- Basic mathematical constructions using the Axioms: Ordered pairs, relations and functions, Cartesian product, partial- and linear-order relations, equivalence relations.
- Natural numbers. The Axiom of Infinity and the set-theoretic definition of the natural numbers.
- Cardinalities. Basic operation on cardinalities. Elementary properties of cardinal numbers. Cantor-Bernstein 'Sandwich' Theorem and its consequences, |A| < |P(A)|.
- Well-orderings. Transfinite Induction and Recursion : Ordinal numbers, and ordinal arithmetic.
- Axion of Choice and its equivalents : the Well Ordering Theorem, Zorn lemma, and the Fundamental Theorem of Cardinal Arithmetic.
- Applications (as many as time permits):
- Hamel basis; the additive groups of the reals and of the complex numbers are isomorphic; the function f(x)=x is the sum of two periodic functions.
- Mazurkiewicz theorem: there is a subset of the plain which intersects every line in exactly two points. Find/create generalizations of this theorem
- Dehn's Theorem about decompositions of geometric bodies
- Sierpinski's Theorem and the Continuum Hypothesis,
- decomposition of R3 into congruent circles,
- Infinite combinatorics: pressing-down lemma, partition theorems, Δ-systems
- Goodstein's Theorem.
- A glimpse of independence proofs: New axioms: large cardinals, ⋄ and Martin's Axiom
Books:
- Karel Hrbacek, Thomas Jech: Introduction to Set Theory, (Chapman & Hall/CRC Pure and Applied Mathematics)
- Yiannis Moschovakis: Notes on Set Theory (Undergraduate Texts in Mathematics, Springer)
Grading: Homework assignments: 50%, midterm exam: 20%, final exam: 30%.
A: 80-100%, B: 70-79%, C: 60-69%, D: 50-59%