Instructor: Dr. Lajos SOUKUP
Website of the course: http://www.renyi.hu/~soukup/set_14f.html
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 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 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.
A: 80-100%, B: 70-79%, C: 60-69%, D: 50-59%
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:
- Extensionality: Two sets are equal if and only if they have the same elements.
- 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. 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: x^{2}= 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 R^{3} 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?