Abstract: We continue with the topic of the previous week. We will define the Radin forcing, discuss (without proof) and its basic properties. We will give Woodin's proof for the consistency of the existence of strong inaccessible without diamond and show how to strengthen it to the consistency of strong inaccessible without weak diamond.

This talk is about three published papers of mine that form my phd. In the first two chapters I focus in the model theory of real closed fields and in the third one I take one step back and investigate in greater genearility dependent theories. The results are the following: 1. Boundedness criterion for rational functions over generalized semi-algebraic sets in real closed fields. 2. Positivity criterion for polynomials over generalized semi-algebraic sets in real closed valued fields.

Sela proved that the theory of free groups is stable. It is thus natural to wonder what the independence relation looks like. Together with Sklinos, we worked out a complete characterization of forking independence in the standard model (over any set of parameters) in terms of an algebraic-geometric object called the JSJ decomposition, which encodes all the splittings of the group as an amalgamated product or an HNN extension relative to the parameter set. In the talk we will try to give an idea of the proof over some examples.

A special class among the countably infinite relational structures is the class of homogeneous structures. These are the structures where every finite partial isomorphism extends to a total automorphism. A countable set, the ordered rationals, and the random graph are all homogeneous.

We isolate the property of being a critical point, and prove some basic positive properties of them. We will also prove a lifting property that allows lifting elementary embedding to symmetric extensions, and outline a construction that shows that it is consistent that a successor of a critical cardinal is singular. This is a recent work with Yair Hayut.

Abstract: The notion of an ascent path through a tree, isolated by Laver, is a generalization of the notion of a cofinal branch and, in many cases, the existence of an ascent path through a tree provides a concrete obstruction to the tree being special. We will discuss some recent results regarding ascent paths through kappa-trees, where kappa > omega_1 is a regular cardinal. We will discuss the consistency of the existence or non-existence of a special mu^+-tree with a cf(mu)-ascent path, where mu is a singular cardinal.

We will take a close look at the first few steps of the construction of the Bristol model, which is a model intermediate to L[c], for a Cohen real c, satisfying V eq L(x) for all x.

We shall try to prove some surprising (and hopefully, correct) theorems about the relationship between the club principle (Hebrew: tiltan) and the splitting number, with respect to the classical s at omega and the generalized s at supercompact cardinals.