# MAT 4930 and MHF 6306

Course: MAT 4930 section 0402, course number 24145; MHF 6306 section 3620, course number 16957

Meets: MWF 6th period, 12:50pm-1:40pm, LIT 423

Instructor: Jindrich Zapletal office hours MWF 5th, in LIT 456 and also on zoom The email address is zapletal@ufl.edu

Course prerequisites: The students should have attended a standard undergraduate set theory course equivalent to the UF course MHF 5107. If not, please talk to the instructor to find out if your background is suitable.

Course content: This is the first, set theory part of a one year graduate sequence in mathematical logic. The two parts of the sequence are quite independent of each other and can be taken in any order. The course is divided into four blocks, the dates are somewhat aspirational:

1. Axiomatization of set theory, transfinite recursion and induction, transitive models of set theory. This part mostly gathers theorems and facts of basic ZFC set theory that most students should be to some extent familiar with. Aug. 23—Sep. 3
2. Goedel’s constructible universe. This is the model of set theory which Goedel found in the 1930’s and used it to show that the Continuum Hypothesis cannot be disproved from the axioms of ZFC set theory. Goedel’s proof is our main goal. If time permits, I will introduce variations of this model and also the HOD model. Sep. 8—Oct. 1
3. Cohen’s method of forcing. This is the method that Cohen used in 1960’s to show that the Continuum Hypothesis cannot be proved from the axioms of ZFC, and the axiom of choice cannot be proved from the axioms of ZF. I will introduce the basics of forcing, the forcing theorem, and prove the main two Cohen’s results. Oct 4—Nov. 5
4. Descriptive set theory. This is the science of Polish spaces (topological spaces similar to the reals) and their definable subsets (such as Borel or analytic sets). My goal is to discuss basic constructions of Polish spaces, and then prove Suslin’s theorem: a subset of a Polish space is Borel if and only if both it and its complement are analytic sets. Nov. 8—Dec.8

Textbook, recommended reading: This depends on the block. Schindler: Set Theory, Springer Verlag 2014, ISBN 9783319067247 available in the UF library as an e-book under call number QA248.S319 2014 follows the outline of our course very closely, and it is a great recent book. There are other options, some of them more classical, as listed below.

1. An undergraduate treatment is provided in the book Set Theory and Foundations of Mathematics. Volume 1: Set Theory, which is coauthored by Doug Cenzer, Jean Larson, Chris Porter, and Jindrich Zapletal; it is available as an e-book in the UF library under call number QA248.C358 2020. A more condensed treatment is in Thomas Jech: Set Theory, Section I.1, which is available as an e-book under call number QA248.J42 2003eb. Many other options are perfectly fine.
2. Jech: Set Theory II.13 or Kunen: Set Theory, An introduction to Independence Proofs, Chapter VI call number QA248.K75
3. There are several equivalent treatments of forcing which on the surface may seem very different. I will avoid the Boolean-valued model approach of Jech, and use the more direct combinatorial approach of Kunen or Schindler.
4. I hope to motivate as many people as possible to look into the standard textbook, Kechris: Classical Descriptive Set Theory, UF library e-book under the call number QA248.K387 1995. This covers a lot of stuff we cannot possibly go into, but it is a classic, greatly superior to any other treatment of the subject.

Grading, attendance:  Attendance will not be taken. There will be four equally weighted homework assignments, one per block. I do not want to commit to precise due dates at this point. The students will get a week to work on each, the due date of the last one will fall into the exam week. The homework assignments will consist of proofs. At the end, the letter grade will be calculated from the homework grades using the standard curve: A=93% and higher, A-=90-93%, B+=87%-90% and so on. There is no final exam. More information on UF grading policy may be found at:  UF Graduate Catalog and Grades and Grading Policies

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