## Prof. Dr. Moritz Weber

## Set theory and forcing

(Summer term 2019)### News

For the oral exams, please write an email to Moritz Weber and fix a date.See below for some information on the material covered in the exams.

The following dates are proposed (but individual dates are also possible):

- Wed, 17 July, 8:30 - Marc
- Wed, 17 July, 9:00 - Mirko

- Mon, 16 Sept, 10:30 - Simone
- Mon, 16 Sept, 11:00 - Eileen
- Mon, 16 Sept, 11:30 - Steven
- Mon, 16 Sept, 12:00 - Lisa
- Mon, 16 Sept, 14:00 - Moritz K.
- Mon, 16 Sept, 14:30 - Cedric
- Mon, 16 Sept, 15:00 - Simon
- Mon, 16 Sept, 15:30 - Niklas
- Mon, 16 Sept, 16:00 - Julien

### Time and place of the lecture

Tuesday, 14:00-15:30, HS II (building E2 5)*Schedule:*

- 9 April: Introduction and overview

Introduction

- 16 April: Primitive recursive functions and predicates; Gödel numbering/coding

Ch. 01 - 23 April: Partially recursive functions and the arithmetic hierarchy

Ch. 02 - 30 April: Formal languages; the word problem is undecidable

Ch. 03 - 7 May: PL1 (first order logic)

Ch. 04 - 14 May: Logical calculus; model theory

Ch. 05, Ch. 06 **21 May: Gödel's incompleteness theorems (Gödel 1931)**

This special lecture is an embedded talk - it is designed in a self-contained way;

in principle no attendence of the previous lectures is required (modulo details of the talks)

Ch. 07

- 28 May: The axioms ZF; ordinal numbers

Ch. 08, Ch. 09 - 4 June: The axiom of choice; cardinal numbers

Ch. 10 - 11 June: Inner models; ZF + V=L implies (AC) and (CH) (Gödel 1938)

Ch. 11 **25 June: The Forcing Theorem and ZFC independence of the Continuum Hypothesis**(Cohen 1963)

This special lecture is an embedded talk - it is designed in a self-contained way;

in principle no attendence of the previous lectures is required (modulo details of the talks)

Ch. 12**9 July: Definability in Mathematics, Philipp Lücke (lecture hall IV!)**

Research talk on a topic related to the lecture.- 16 July: no lecture

### Exercise sessions

Tuesday, 15:45-17:15, SR6 (room 217, building E2 4)### Contents

Set theory is the attempt to axiomatize mathematics and to put it on a solid basis; this system of axioms is usually called ZFC (Zermelo-Fraenkel plus the aciom of choice). It is a central theory in mathematical logic. The main issue is to define what a set actually is, or in other words, to answer the question: Which sets are allowed to be sets? Forcing is a very powerful tool from set theory with which we may "force" certain sets to be sets. With the help of forcing, we can create a universe of sets in which ZFC and certain additional axioms or statements hold true. A famous example of the use of forcing is the proof that the Continuum Hypothesis (CH) is independent from ZFC (recall that the Continuum Hypothesis roughly states that the cardinality of the power set of the natural numbers is the next biggest cardinality after the natural numbers): There is a universe of sets satisfying ZFC and CH; but there is also another universe of sets satisfying ZFC and the negation of CH. We conclude: We may not deduce CH from ZFC; hence any attempt to "prove" CH (or to disprove it) must necessarily fail. In the meantime, many other very concrete statements from nowadays mathematics have been proven to be independent from ZFC (like the existence of outer automorphism of the Calkin algebra). So, before trying to prove a very hard problem in mathematics, we might better check whether it is independent from ZFC. The lecture is held in English. This lecture basically does not require any mathematical background and is somewhat independent from the usual curriculum. Note however, that some of the proofs will be only sketched if not omitted and parts of the lecture might require some background reading by the participants. This style of a lecture is probably quite ambitious for students from the first year. |

### Oral exams

Here are some questions and topics for the oral exams (20-30 min) in set theory:- what are primitive recursive terms, what are primitive recursive functions, what are partially

recursive functions, what are the links, small examples - what are primitive recursive predicates, recursive predicates, recursively enumerable predicates,

links, small examples, what is the arithmetic hierarchy (Def 2.23) - what is the idea behind Gödel coding and Gödel numbers
- what is the idea of universal functions and predicates as in Thm 1.20 and Thm 2.13, 2.14
- what is a first order language, what are formulas and theorems and interpretations in the sense

of Sect 4. - how to formalise “a theorem may be deduced from a theory”, Def. 4.7 and 5.1, links
- what is (very roughly) the statement of Gödel’s incompleteness theorems (and possibly, what is a

very rough idea of the proof) - Name and explain some axioms of ZFC, what is the main issue with the definition of a set, what

is a class - you may leave out Sect. 3 and 6. From Sect 9-12 only very rudimentary questions might be asked

(like, what is an ordinal, what is a cardinal), in exceptional cases

### References

- Kunen, Set Theory, 2011
- Jech, Set Theory, 1978
- Schindler, Set Theory: Exploring independence and truth, 2014
- Weaver, Forcing for mathematicians, 2014

Link to the lecture Set Theory and Logic by Gunter Fuchs including some lecture notes

Last update: 13 Sept 2019 Moritz Weber | Impressum |