HUJI Logic Seminar

** Tuesday 29/5 13:30-15:00 Ross 63**

__Title:__ Higher Random Reals

__Speaker:__ Martin Goldstern

__Abstract:__

The set of real numbers is often identified with

Cantor Space 2^omega, with which it shares many important

properties: not only the cardinality, but also other

“cardinal characteristics” such as cov(null), the smallest

number of measure zero sets needed to cover the whole space,

and similarly cov(meager), where meager=”first category”;

or their “dual” versions non(meager) (the smallest

cardinality of a nonmeager set) and non(null).

Many ZFC results and consistency results (such as

“cov(meager) lessequal non(null), but no inequality between

cov(null) and cov(meager) is provable in ZFC”)

are known.

Recent years have seen a renewed interest in “higher reals”,

i.e., elements of 2^kappa, where kappa is usually an inaccessible

cardinal. Meager sets have a natural generalisation to this

context, namely “kappa-meager” sets (using the <kappa-box product

topology), but what is the natural generalisation of the

ideal of null sets?

In my talk I will present an ideal null_kappa recently introduced

by Saharon Shelah, and some ZFC and consistency results from a

forthcoming joint paper with Thomas Baumhauer and Saharon Shelah,

such as “cov(null_kappa) lessequal non(null_kappa)”, and

“consistently, cov(meager_kappa) > cov(null_kappa)”.