# Midterm: Wednesday, Oct 24, 10.00 -- Friday, Oct 26, 10.15

You can dowload the Midterm Instructor: Dr. Lajos SOUKUP

E-Mail: If you have any question, do not hesitate to write me: soukup@renyi.hu or lsoukup@gmail.com

Gmail chat: lsoukup@gmail.com

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

Classes: Wednesday 1015-11, 1115-12, Friday 1015-11, 1115-12, Room 104

Office Hour (Tutorial): Wednesday 1115-12 from the second week

Text: The course is based on notes and printed handouts, which are distributed usually on Friday

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 learn how to use set theory as a powerful tool in algebra, analysis, and even geometry,
• we get an insight how set theory can serve as the foundation of mathematics,
• 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 on Friday and collected on Wednesday

Topics:

• Classical set theory. Cantor: "By a set we are to understand any collection onto a whole of definite and separate objects of out intuition or our thought."
• Basic operations on sets. 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, 2ω = c (the cardinality of the real line), there are c many continuous functions, 1· 2 · 3 ··· = c, the cardinal numbers c, 2c, etc., Kőnig'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: x2= x for every cardinal x.
• Applications (as many as time permits):
• The fall of 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.
Does the set of all those sets that do not contain themselves contain itself?
'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.

Tentative Course Calendar
• Week 1. Introduction. Countable sets.
• Week 2. Combinatorics of countable sets.
• Week 3. Cardinalities
• Week 4. Operation on cardinalities
• Week 5. Well-ordered sets. Axiom of Choice and Zorn lemma
• Week 6. Transfinite induction and recursion
• Week 7. Midterm. Well-order trichotomy
• Week 8. Ordinals
• Week 9. Cardinals, alephs, cofinalities.
• Week 10. Ordinal arithmetic with applications
• Week 11. Axiomatic set theory
• Week 12. Infinite combinatorics
• Week 13. Selected problems
• Week 14. Final