Choice and Regularity: Common Consequences in Logic

TL;DR

This paper explores the intersection of Choice and Regularity axioms in set theory, introducing a new principle called Best-Foundedness that captures their common consequences and analyzing their logical relationships and implications.

Contribution

It introduces the Best-Foundedness principle as a unifying concept linking Choice and Regularity, and studies its logical properties and consequences.

Findings

Best-Foundedness implies key consequences of Choice and Regularity.

It is consistent with the negations of both Choice and Regularity.

The paper analyzes interpretability strength and related set-theoretic principles.

Abstract

It is well-known that Choice and Regularity are independent of each other but have important common consequences of logical character (reflection principles, representations of classes by sets, etc.). We explain this phenomenon by isolating their "intersection", a principle (called here Best-Foundedness) which is consistent with the negations of both axioms but implies all these consequences. Then we study relationships between these consequences (and near principles) in detail. Finally, we consider some arguments related to truth of various principles in set theory, especially arguments concerning the interpretability strength.