Biba's trick, with applications

TL;DR

This paper advances the understanding of forcing axioms in set theory by improving and simplifying key lifting theorems related to quotient structures and extending their applicability to broader classes of ideals.

Contribution

It provides a simplified proof of the OCA lifting theorem and extends the 2004 lifting theorem to a wider class of ideals with weaker assumptions.

Findings

Improved proof of the OCA lifting theorem

Extended lifting theorem to countably 80-determined ideals

Weakened assumptions for lifting results

Abstract

We give another bit of evidence that forcing axioms provide proper framework for rigidity of quotient structures, by improving the OCA lifting theorem proved by the author in late 20th century and greatly simplifying its proof. In the assumptions of this theorem. We also extend the conclusion of author's 2004 lifting theorem from a lifting result for countably 3204-determined ideals to one for countably 80-determined ideals and weaken its assumptions.