2020: Large Cardinals

Here are the recorded lectures from the Mastermath course in Set Theory (autumn 2020) about Large Cardinals and some preparatory lectures on cardinal arithmetic, combinatorics, stationary sets, and (ultra)filters.

2020-10-26: Cardinal Arithmetic

What can be proved about the continuum function and cardinal exponentiation by relatively elementary means.
the writings and the homework

2020-11-02: Filters, ultrafilters and stationary sets

• Filters and ultrafilters; the ultrafilter theorem.
• Ramsey ultrafilters on \(\omega\).
• Completeness of filters: the question about \(\sigma\)-complete ultrafilters foreshadows measurable cardinals.
• Closed unboundes sets, stationary sets and Fodor's Pressing-Down Lemma.

the writings and the homework

2020-11-09: Combinatorics

Ramsey's Theorem, selective ultrafilters have the Ramsey property, The Erdős-Rado theorem; limiting examples.
the writings and the homework

2020-11-16: More combinatorics

Erdős-Dushnik-Miller theorem, trees, Kőnig's Infinity Lemma and Aronszajn trees. The tree property.
the writings and the homework

2020-11-23: Large Cardinals I

What are large cardinals anyway? Weak compactness and the tree property. Elementary properties of measurable cardinals.
the writings and the homework

2020-11-30: Large Cardinals II

Ultrapowers of the Universe. Definition of ultrapowers; well-foundedness from \(\sigma\)-complete ultrafilters. The Mostowski collapse of the ultrapower by a \(\kappa\)-complete ultrafilter on \(\kappa\) and a few useful properties. Proof using ultrapowers that there are stationarily many weakly compact cardinals below a measurable cardinal.
the writings and the homework

2020-12-07: Large Cardinals III

Why weakly compact? Finished up on Ultrapowers of the Universe. Discussion of languages of type \(\mathcal{L}_{\kappa,\omega}\) and \(\mathcal{L}_{\kappa,\kappa}\) and the Weak Compactness Theorem for these. That theorem holds for weakly compact cardinals, hence their name.
the writings and the homework

2020-11-14: Large Cardinals IV (forgot to record the second hour …)

Indescribability. Finished up the proof of Weak Compactness for weakly compact cardinals. We define \(\Pi^1_1\)-indescribability and show that \(\kappa\) is weakly compact iff it is \(\Pi^1_1\)-indescribable. Applications of this: stationary subsets reflect and there are many inaccessible, Mahlo, … cardinals below a weakly compact cardinal.
the writings and no homework