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:
- Extensionality : Two sets are equal if and only if they have the same elements.
- 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 R^{3} 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
- 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.