Set Theory — SET

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.

In this course, set theory is developed axiomatically, but the treatment is not formal. Logical apparatus is kept to minimal, and logic formalism is completely avoided.

Topics:

• Naive set theory. Basic principles:
  1. Extensionality: Two sets are equal if and only if they have the same elements.
  2. 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

Text: The course is based on printed handouts, which are distributed after classes.

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%