Logic Seminar - Omer Ben-Neria - "Singular Stationarity and Set Theoretic Generalizations of Algebras"

Wed, 27/12/201711:00-13:00
Ross 63
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
Jonsson cardinals).
In the 1990s Foreman and Magidor used algebras to initiate a study
of Singular Stationarity, i.e., a study of alternative notions of
stationarity for subsets of singular cardinals. They introduced and
developed two notions of singular stationarity called Mutual
Stationarity and Tight Stationarity, and used their findings to prove a
fundamental result concerning generalized nonstationary ideals.
The two notions of singular stationarity have been studied in the last
two decades, and the main purpose of the talk to describe the related
known and recent results.
I will start by giving some background material on algebras and
stationary sets, and describe the history of a well-known problem by
Jonsson. We will then proceed to describe the work done on the notion of
Mutually Stationary Sequences and sketch a recent proof which is based
on the existence of special types of strong filters.
In the second part of the talk, we will connect the notions of Singular
Stationarity to results from PCF theory and Extender-based forcing