BSM Set Theory — SET, Fall 2006

Course description: The goal of the course is twofold. On the one hand, we get an insight how set theory can serve as the foundation of mathematics, and on the other hand, we learn how to use set theory as a powerful tool in algebra, analysis, geometry and even number theory.

Grading: Homework assignments: 20%, midterm exam: 30%, final exam: 50%
A: 80-100%, B: 60-79%, C: 40-59%, D: 20-39%

Syllabus:

• Introduction: Notation, empty set, union, intersection, complement, subset, power set, equality of sets, N, Z, Q, R, countable and uncountable sets.
• Elementary properties of cardinal numbers: Equivalence of sets, cardinals, the Cantor-Bernstein 'Sandwich' Theorem and its consequences, |A| < |P(A)|.
• 'Oops': Russel's Paradox.
• The axiomatic approach: Zermelo-Fraenkel Axioms.
• More on cardinal numbers: Calculations with cardinals, 2^ω = c (the cardinality of the continuum), there are c many continuous functions, 1· 2 · 3 ··· = c, the cardinal numbers ω, 2^ω, 2^2ω, etc., König's Inequality.
• Ordered sets: Definition, isomorphism, initial segment, initial segment determined by an element, order type, well ordered set.
• The crucial notion, 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: &kappa²=&kappa for every infinite cardinal κ, Zorn's lemma.
• Applications (as many as time permits): Every vector space has a basis, Hamel basis^*, Cauchy's Equation, Dehn's Theorem about decompositions of geometric bodies, the Long Line, f(x)=x is the sum of two periodic functions, 2-point sets, Sierpinski's Theorem^* and the Continuum Hypothesis, throwing darts at the plane, decomposition of R³ into circles, Goodstein's Theorem^*, the Problem of 13 Numbers, nonstandard analysis (infinitesimal numbers and how to make dy/dx precise), how to define the limit of nonconvergent sequences (Banach limits and ultrafilters).