Algebra Seminar talk
2019-10-11
Martin Goldstern
Uncountable cardinals in Analysis and Cichoń's Diagram
Abstract:
Cichoń's Diagram describes a partial order between 10 uncountable cardinals, among them
- cov(null) = the smallest number of Lebesgue null sets needed to cover all real numbers,
- non(meager) = the smallest cardinality of a non-meager set (=set of second category),
- as well as aleph1 (smallest uncountable cardinal)
- and c (the cardinality of R, the set of all real numbers).
In 1963, Paul Cohen invented the method of "forcing" and used it to show the unprovability of Cantor's Continuum Hypothesis, or in other words: that the continuum does not necessarily have cardinality aleph1.
It is still open which values the other cardinal's in Cichoń's Diagram may take, but in a recent paper (with Jakob Kellner and Saharon Shelah) we could for the first time construct a set-theoretic universe in which all cardinals in Cichoń's Diagram have different values.
I will talk a bit about the cardinals in Cichoń's Diagram and hint at the methods which allow us to control/manipulate their values.
This talk is aimed at a general mathematical audience, with no prior knowledge of ZFC-models or forcing.
If there is sufficient interest, there will be a "Part 2" on October 18th.