BSM Set Theory — SET, Fall 2008.

Final: Wednesday, Dec 17, 2008

This is an "open notes" exam

Instructor: Dr. Lajos SOUKUP

E-Mail: If you have any questions, do not hesitate to write me:

Homepage: http:www.renyi.hu/~soukup

Classes: Tuesday 10¹⁵-11, 11¹⁵-12, Wednesday 10¹⁵-11, 11¹⁵-12, Lecture Hall 206

Office Hour (Tutorial) Tuesday 11¹⁵-12 from the second week.

Text: The course is based on handouts

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

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

Homeworks are distributed and collected on Wednesday

Topics:

• Naive set theory: general principle of comprehension, due to Frege (1893): If P is a property then there is a set Y={X:P(X)} of all elements having property P.
• Contradictions in mathematics?
  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.
• The crucial notion of "well-ordering", ordinal numbers: Definition, properties, calculations with ordinals.
• Elementary properties of cardinal numbers: Equivalence of sets, cardinals, the Cantor-Bernstein 'Sandwich' Theorem and its consequences, |A| < |P(A)|.
• 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 c, 2^c, etc., König's Inequality.
• 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, Cauchy's Functional Equation, Dehn's Theorem about decompositions of geometric bodies, the Long Line, f(x)=x is the sum of two periodic functions, Sierpinski's Theorem and the Continuum Hypothesis, decomposition of R³ into circles, Goodstein's Theorem^*.