Kurt Gödel Research Center

2012 seminar talk: Cardinal characteristics in Cichon's diagram

Talk held by Martin Goldstern (TU Wien, Austria) at the KGRC seminar on 2012-11-22.

Abstract

Cichon's diagram describes the relationships between 12 infinite cardinal numbers, among them

• $\aleph_1$ and $\mathfrak{c}$=continuum (the smallest and largest of the 12)
• $\operatorname{cov(N)}$ and $\operatorname{cov(M)}$, the covering numbers of the ideals of Lebesgue null and meager sets, respectively i.e., the answers to the questions "how many null/meager sets do we need to cover the reals?"
• $\mathfrak{b}$ and $\mathfrak{d}$, the bounding and dominating numbers. (The dominating number $\mathfrak{d}$ is the smallest size of a family of $\sigma$-compact sets covering the irrationals.)

"ZFC + $\mathfrak{c}=\aleph_1$" implies of course that all 12 cardinals have the same value. All possible consequences for Cichon's diagram of the axioms "ZFC + $\mathfrak{c}=\aleph_2$" are known, or in other words: for every assignment of the values $\aleph_1$ and $\aleph_2$ to the cardinals in Cichon's diagram it is known whether there is a ZFC-universe in which this assignment is realized.

It is notoriously difficult to construct universes with prescribed properties in which $\mathfrak{c}$ is large (even just larger than $\aleph_2$).

After discussing background and known results, I will highlight some features of a construction that will appear in a joint paper with Arthur Fischer, Jakob Kellner and Saharon Shelah. Our construction is a "creature iteration" (which is almost, but not quite, entirely unlike a product of creature forcings). All universes we construct will satisfy $\mathfrak{d}=\aleph_1$ (which implies, among others, $\operatorname{cov(M)}=\aleph_1$), while the cardinals that are not obviously bounded by $\mathfrak{d}$ can have (almost) arbitrary regular values.