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Logic Seminar - Daniel Iosub | Einstein Institute of Mathematics

Logic Seminar - Daniel Iosub

Date: 
Wed, 23/03/202211:00-13:00
Location: 
Via zoom: Meeting ID: 891 8649 9242 , Passcode: 360371
Title: Consistency results related to the singular cardinal hypothesis and PCF theory

Abstract:


The singular cardinal hypothesis (SCH) states that if $\kappa$ is a strong limit cardinal then $2^\kappa = \kappa^+$. While SCH is consistent relative to ZFC, the negation of SCH is strictly stronger - we must assume the existence of some large cardinals if we want to produce a model where SCH fails.
The first purpose of this talk is to give a brief introduction to the history of the problem, and to the methods used to show that the negation of SCH is consistent.

Another question we can ask in the context of SCH, is how bad can it fail? That is, if $\kappa$ is a strong limit cardinal then how large can $2^\kappa$ be? For example, by a famous result due to Saharon Shelah, if $\aleph_\omega$ is a strong limit then $2^{\aleph_\omega} < \aleph_{\omega_4}$. This result is obtained by using Shelah's PCF theory (PCF stands for "possible cofinalities").

The second purpose of this talk is to introduce the basic ideas of PCF theory and discuss the relation between it and the singular cardinal hypothesis. I will finish the talk by briefly discussing results from my ms.c thesis, in which I study ways to obtain a certain PCF configuration below $\aleph_\omega$.

I will only assume basic knowledge of set theory. Some of the topics of the talk are more involved, but I will try to keep most of the discussion at an informal and an intuitive level.