## Seminar Mathematical Logic

### 2014/2015; 1st Semester; Block A

**Institute for Logic, Language & Computation**

**Universiteit van Amsterdam**

**Instructor:** Prof. Dr. Benedikt Löwe, with some assistance by Hugo Nobrega.

**Course language:** English

**Goal of this course:** Students need to learn to read research papers and present mathematical ideas. The goal of this seminar is to give students an opportunity to train their reading skills in mathematics and learn how to present.

**Content of the course:** Highlights of the foundations of mathematics

**Evaluation:**
There will be no number grades, just AVV (has met all requirements, "passed") or NAV (hasn't met all requirements, "failed"). Necessary condition to pass the course is to give a talk.

**Schedule and contents of talks.**

* Note:* Due to several scheduling conflicts, the time and place of many talks differs to the one found on DataNose – in particular all talks with the exception of the one on October 3 will take place in the seminar room of the ILLC (room F1.15 of Science Park 107).
The actual schedule for the seminar is as follows.

22 September 2014 Monday |
13-15 SP107 F1.15 |
Undefinability of truth
(Stephen Pastan).
In this talk, we'll prove Tarski's theorem on the undefinability of truth. We will also discuss a strengthening of this result: we introduce a complexity measure on definable concepts (quantifier complexity) and show that the set of true Pi_1 formulas cannot be Sigma_1. We observe that the set of provable Pi_1 formulas is Sigma_1, and derive Goedel's first incompleteness theorem as an immediate corollary: there is a true but unprovable Pi_1 sentence. |

23 September 2014 Tuesday |
17-19 SP107 F1.15 |
Goedel's second incompleteness theorem
(Kristina Gogoladze)
In the foundations of mathematics, Goedel's 2nd incompleteness theorem is more important than his first. Instead of just providing *some* true but unprovable statement, the 2nd incompleteness theorem provides a concrete example: the consistency statement of the theory in question. |

29 September 2014 Monday |
13-15 SP107 F1.15 |
Independence proofs in basic axiomatic set theory, part 1
(Ur Ben-Ari-Tishler).
We compare Zermelo set theory Z and Zermelo Fraenkel set theory ZF and prove that ZF is strictly stronger than Z by proving that ZF proves the consistency of Z. |

1 October 2014 Wednesday |
13-15 SP107 F1.15 |
Independence proofs in basic axiomatic set theory, part 2
(Fangzhou Zhai).
One of the fundamental questions of axiomatic set theory is the question whether there are regular limit cardinals. We look at a slight strengthening, regular strong limit cardinals, also called *inaccessible cardinals* and show that ZF does not prove their existence by using the techniques developed in the fifth talk. |

3 October 2014 Friday |
13-15 SP904 G3.05 |
The consistency of the axiom of choice
(Eiseart Dunne).
But maybe the axiom of choice is inconsistent after all? No, using the notion of definability, we can construct a model of ZFC inside each model of ZF: the class of hereditarily ordinal definable sets. |

6 October 2014 Monday |
13-15 SP107 F1.15 |
The axiom of choice and bases of vector spaces
(Frederik Lauridsen).
The most well-known example of a use of the axiom of choice is the proof that every vector space has a basis. In fact, this statement itself is equivalent to the axiom of choice, and in this talk we'll look at the equivalence proof. |

7 October 2014 Tuesday |
17-19 SP107 F1.15 |
An axiom in conflict with the axiom of choice
(Mike Hirsch).
In the 1960s, Jan Mycielski introduced an axiom called "The Axiom of Determinacy" (back then called "Determinateness") and proposed it as an alternative to the axiom of choice. In this talk, we shall introduce this axiom (a game-theoretic axiom) and prove that it contradicts the axiom of choice. |

8 October 2014 Wednesday |
13-15 SP107 F1.15 |
Lebesgue measure, choice and determinacy
(Sarah Hiller).
The axiom of choice implies that there are non-Lebesgue measurable sets. However, the axiom of determinacy implies that all sets are Lebesgue measurable. |

10 October 2014 Friday |
13-15 SP107 F1.15 |
The Banach-Tarski paradox
(Olim Tuyt).
Continuing with the axiom of choice: one of the most geometrically puzzling results in set theory is the fact that you can take apart a three-dimensional solid ball using volume-preserving maps, reassemble it, and produce two three-dimensional solid balls of the same volume. |

13 October 2014 Monday |
13-15 SP107 F1.15 |
The consistency of the negation of the axiom of choice
(Zeno de Hoop).
The axiom of choice cannot be proved. For this talk, we leave the setting of ZF and move to ZFU (ZF set theory with urelements) and use a technique called Fraenkel-Mostowski models to produce a model of ZFU + not-AC. |

15 October 2014 Wednesday |
13-15 SP107 F1.15 |
The consistency of the continuum hypothesis
(Mees de Vries).
We move to one of the most famous open questions of set theory: the question whether there is a bijection between aleph_1 and the set of real numbers. In 1938, Goedel famously proved that this statement cannot be disproved by constructing a model called the "constructible universe" and proving the "condensation lemma" about it. |

17 October 2014 Friday |
13-15 SP107 F1.15 |
Suslin's problem
(Tingxiang Zou).
The real line is uniquely characterized as the dense complete linear order without endpoints that has a countable dense subset. What if you replace this last property by "every collection of mutually disjoint open intervals is countable" (the so-called "countable chain condition")? Suslin asked whether a characterization theorem can be proved? A linear order that gives a negative answer to Suslin's question is called a Suslin line. Jensen proved that the diamond principle implies that Suslin lines exist. |