Martin Goldstern: Higher Random Reals

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)”.

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