On Friedman's Property

TL;DR

This paper introduces new forcing techniques that add witnesses to failures of Friedman's Property while preserving large cardinal features, enabling separation of various problem variants.

Contribution

It defines novel forcing orders that facilitate the construction of master conditions, advancing the understanding of Friedman's Property and related set-theoretic principles.

Findings

Separated different variants of Friedman's Problem at the same and different cardinals.

Established new results on the differences between <κ- and κ-strategic closure.

Demonstrated preservation of large cardinal properties using the new forcing techniques.

Abstract

We define forcing orders which add witnesses to the failure of various forms of Friedman's Property. These posets behave similarly to the forcing order adding a nonreflecting stationary set but have the advantage of allowing the construction of master conditions and thus the preservation of various large cardinal properties. We apply these new techniques to separate various instances of variants of Friedman's Problem, both between different instances at one cardinal as well as equal instances at different cardinals and en passant obtain some new results regarding the differences between ${<}\,\kappa$- and $\kappa$-strategic closure.