This talk will be largely based on a paper by Joseph Shipman with the same title. We will discuss some variations of Fubini type theorems. The focus will be on what is known as "strong Fubini type theorems". Apparently these versions were proved to be independent of ZFC,and our main aim will be to sketch a proof of this result. We will assume basic knowledge in measure theory. Aside from that, the material is rather self contained.
Abstract: The set theoretic generalizations of algebras have been
introduced in the 1960s to give a set theoretic interpretation of usual
algebraic structures. The shift in perspective from algebra to set
theory is that in set theory the focus is on the collection of possible
algebras and sub-algebras on specific cardinals rather than on
particular algebraic structures. The study of collections of algebras
and sub-algebras has generated many well-known problems in combinatorial
set theory (e.g., Chang’s conjecture and the existence of small singular
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.
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.
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.
Chang's Conjecture is a strengthening of Lowenheim-Skolem-Tarski theorem. While Lowenheim-Skolem-Tarski theorem is provable in ZFC, any instance of Chang's Conjecture is independent with ZFC and has nontrivial consistency strength. Thus, the question of how many instances of Chang's Conjecture can consistently hold simultaneously is natural.
I will talk about some classical results on the impossibility of some instances of Chang's Conjecture and present some results from a joint work with Monroe Eskew.
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.