Set Theory — SET

Instructor: Dr. Lajos SOUKUP
Website of the course: http://www.renyi.hu/~soukup/set_15f.php

Text: A. Shen, and N. K. Vereshchagin, Basic Set Theory, AMS Student Mathematical Library 17, and printed handouts (distributed after classes)

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

Grading:
Homework assignments: 40%, midterm exam: 20%, final exam: 40%.
12 homework assigments, the best 10 count.
Extra hw problems for extra credits.
A: 80-100%, B: 60-79%, C: 50-59%, D: 40-49%

Course description
Set Theory is the study of infinity.

• We build the rich mathematical theory of different sizes of infinity with Cantor's definitions of cardinal and ordinal numbers;
• we learn how to use set theory as a powerful tool in algebra, analysis, combinatorics, and even in geometry;
• we get an insight how set theory can serve as the foundation of mathematics.

Topics:

• Classical set theory: "By a set we are to understand any collection onto a whole of definite and separate objects of out intuition or our thought." (Cantor)

  Basic principles:

  1. Extensionality : Two sets are equal if and only if they have the same elements.
  2. General principle of comprehension of Frege : 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. A sample problem: any family of disjoint letters T on the plain is countable.
• 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!"
• Cardinalities. Comparing the sizes 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, \(2^{\aleph_0}= \mathfrak c\) (the cardinality of the real line), there are \(\mathfrak c\) many continuous functions. Infinite operations on cardinals: \(1\cdot 2 \cdot 3 \cdots = \mathfrak c\). Konig's Inequality.
• The crucial notion of "well-ordering", ordinal numbers: Definition, properties, calculations with ordinals.
• The heart of the matter: The Well Ordering Theorem of Zermelo: we can enumerate everything, the Theorem of Transfinite Induction and Recursion, the Fundamental Theorem of Cardinal Arithmetic: \(\kappa^2=\kappa\) for every cardinal \(\kappa\).
• 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 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" is itself a name consisting of eighteen 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. Natural numbers. Ordinals
• A glimpse of independence proofs: How can you prove that you can not prove something?
• Contemporary Axiomatic Set Theory: extensions of the classical Zermelo-Fraenkel Set Theory with new axioms

Tentative Course Calendar

• Week 1. Sets, cardinalities, equal cardinalities. An application: there is a real number which is not the root of any nonzero polynomial with integer coefficients.
• Week 2. Countable sets, Cantor theorem, Cantor-Bernstein-Schroeder theorem.
• Week 3. Cardinals, operation on cardinals, infinite operations on cardinals.
• Week 4. Equivalence relations and orderings.
• Week 5. Well-founded sets, well-ordered sets.
• Week 6. Transfinite induction and recursion.
• Week 7. Zermelo theorem and its applications. Hamel basis.
• Week 8. Zorn lemma and its applications, "Take Home" Midterm.
• Week 9. Cardinal arithmetic revisited.
• Week 10. Ordinals. Ordinal arithmetic.
• Week 11. Applications of ordinal arithmetic.
• Week 12. Borel sets, cofinality, Mazukievich Theorem.
• Week 13. Axiomatic Set Theory: Mathematical Logic in Nutshell.
• Week 14. Axiomatic Set Theory: Independence Proof. New Axioms. "Take Home" Final.