Continuum Many Different Things: Localisation, Anti-Localisation and Yorioka Ideals

TL;DR

This paper constructs a proper forcing under CH that creates continuum many distinct cardinal characteristics related to Yorioka ideals and localisation concepts, answering several open questions in the field.

Contribution

It introduces a new forcing method that produces continuum many different cardinal characteristics parametrized by reals, expanding understanding of localisation and Yorioka ideals.

Findings

Forcing with $eth_2$-cc adds continuum many distinct cardinals.

Addresses open questions from arXiv:1805.11005.

Demonstrates the diversity of cardinal characteristics under CH.

Abstract

Combining creature forcing approaches from arXiv:1003.3425 and arXiv:1402.0367, we show that, under CH, there is a proper $\omega^\omega$-bounding poset with $\aleph_2$-cc that forces continuum many pairwise different cardinal characteristics, parametrised by reals, for each one of the following six types: uniformity and covering numbers of Yorioka ideals as well as both kinds of localisation and anti-localisation cardinals, respectively. This answers several open questions from arXiv:1805.11005.