Set Theory — SET

Instructor: Dr. Lajos SOUKUP
Homepage of the course: http://www.renyi.hu/~soukup/set_13f.php

Text: The course is based on printed handouts

Books:P. Halmos: Naive Set Theory
P. Hamburger, A. Hajnal: Set Theory
K. Kunen: Set Theory, Chapter 1.
T. Jech: Set Theory, Chapters 1--6.
K. Ciesielski: Set Theory for the Working Mathematician

Prerequisite: Some familiarity with "higher" mathematics. No specific knowledge is expected.

Course description
The goal of the course is threefold:

• we learn how to use set theory as a powerful tool in algebra, analysis, and even geometry,
• we get an insight how set theory can serve as the foundation of mathematics,
• we study how to build up a rich mathematical theory from simple axioms.

Grading: Homework assignments: 40%, midterm exam: 20%, final exam: 40%.
A: 85-100%, B: 70-84%, C: 55-69%, D: 40-54%

Topics:

• Classical set theory. Cantor: "By a set we are to understand any collection onto a whole of definite and separate objects of out intuition or our thought."
• general principle of comprehension (Frege, 1893): If P is a property then there is a set Y={X:P(X)} of all elements having property P.
• Basic operations on sets. Countable and uncountable sets.
• An application: there are uncountably many transcendental real numbers.
• Inductive constructions. A sample problem: "A flea is moving on the integer points of the real line by making identical jumps every seconds. You can check one integer every seconds. Catch the flea!"
• Ramsey Theory. How to prove the finite Ramsey theorem from the infinite one? König lemma: an infinite, locally finite tree should contain infinite paths. Applications: a countable graph is n-colorable if and only if its every finite subgraph is n-colorable.
• Cardinalities. Comparing the size of infinite sets. Cardinalities. Basic operation on cardinalities. Elementary properties of cardinal numbers. Cantor-Bernstein 'Sandwich' Theorem and its consequences, |A| < |P(A)|.
• More on cardinal numbers: Calculations with cardinals, 2^. = c (the cardinality of the real line), there are c many continuous functions, 1ˇ 2 ˇ 3 ˇˇˇ = c, the cardinal numbers c, 2^c, etc., K.nig's Inequality.
• The crucial notion of "well-ordering", ordinal numbers: Definition, properties, calculations with ordinals.
• The heart of the matter: The Well Ordering Theorem: we can enumerate everything, the Theorem of Transfinite Induction and Recursion, the Fundamental Theorem of Cardinal Arithmetic: x²= x for every cardinal x.
• Applications (as many as time permits):
  • Every vector space has a basis; Hamel basis; the additive groups of the reals and of the complex numbers are isomorphic.
  • Mazurkiewicz theorem: there is a subset of the plain which intersects every line in exactly two points
  • Cauchy's Functional Equation: find non-trivial solutions of the function equations f(x)+f(y)=f(x+y),
  • Dehn's Theorem about decompositions of geometric bodies
  • the Long Line
  • the function f(x)=x is the sum of two periodic functions,
  • Sierpinski's Theorem and the Continuum Hypothesis,
  • decomposition of R³ into congruent circles,
  • Goodstein's Theorem.
• Contradictions in mathematics? The fall of naive set theory. The comprehension principle of Frege leads to contradictions.
  • Russel's Paradox: Does the set of all those sets that do not contain themselves contain itself?
  • Berry's Paradox: 'The least integer not nameable in fewer than nineteen syllables'
• The solution: Axiomatic approach (without tears): Mathematical logic in a nutshell. Variables, terms and formulas. The language of set-theory. Zermelo-Fraenkel Axioms.
• Basic Set Theory from the Axioms: Ordered pairs. Basic operations on sets. Relations and functions. Cartesian product. Partial- and linear-order relations.
• A glimpse of independence proofs: How can you prove that you can not prove something?