Many Different Uniformity Numbers of Yorioka Ideals

TL;DR

This paper demonstrates that it is consistent to have uncountably many Yorioka ideals with distinct uniformity numbers and similar diversity in certain cardinal characteristics related to reals, using creature forcing.

Contribution

It introduces a method to produce models where many Yorioka ideals and related cardinal characteristics have pairwise different values.

Findings

Uncountably many Yorioka ideals have pairwise distinct uniformity numbers.

Uncountably many localization and anti-localization cardinals are pairwise different.

The results are achieved using a countable support product of creature forcing posets.

Abstract

Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals' uniformity numbers can be pairwise different. In addition we show that, in the same forcing extension, for two other types of simple cardinal characteristics parametrised by reals (localisation and anti-localisation cardinals), for uncountably many parameters the corresponding cardinals are pairwise different.