250123 VO Special Topics in Set Theory (2022W)

This lecture will devote itself to the classical arithmetic of singular cardinals. We will give an introduction to the basic definitions and results on cardinal arithmetic, the continuum hypothesis and the method of forcing. Our goal is to understand how arithmetic on singulars differs from the arithmetic on regulars, as well as to study classical forcing constructions controlling the continuum function at specific singulars. At the end, we want to give a gently introduction to the theory of possible cofinalities of Shelah. Here a tentative program:

I. Review of basic concepts- cardinality and cofinality.
1. König's Theorem. Exponentiation of cardinals. GCH.
2. A short review on forcing.
3. Easton's theorem.

II. Arithmetic of singular cardinals.
1. The singular cardinal hypothesis.
2. Silver's Theorem.
3. Galvin-Hajnal’s theorems.

III. Large cardinals and the singular cardinals problem.
1. Elementary embeddings and some large cardinal notions.
2. Measurable cardinals and supercompact cardinals.
3. Silver's forcing.
4. Prikry forcing.

IV. Prikry-type forcings.
1. Adding many Prikry-sequences.
2. Nice systems of ultrafilters.
3. Collapsing cardinals.
4. Down to $\aleph_\omega$.

V. A gently introduction on pcf (time availability dependent)