The Slicing Axioms

TL;DR

The paper introduces the family of Slice axioms, which describe specific decompositions of power sets into increasing unions of models, and investigates their consistency with Martin's Axiom, finding limited compatibility.

Contribution

It defines the Slice axioms involving decompositions into models and analyzes their consistency with various forms of Martin's Axiom.

Findings

Slice axioms are compatible only with very weak forms of Martin's Axiom.

The axioms involve decompositions of power sets into increasing unions of models.

Compatibility results restrict the strength of Martin's Axiom with these axioms.

Abstract

We introduce the family of axioms, denoted $\operatorname{Slice}_\kappa$, that claim the existence of strictly increasing decompositions of the form $$2^{\delta}=\bigcup_{\alpha<\kappa} 2^{\delta}\cap M_\alpha,$$ where $\delta<\kappa$, and $\{M_\alpha|\; \alpha<\kappa\}$ is a $\subseteq$-increasing sequence of transitive models of set theory. We study compatibility of these axioms with versions of Martin's Axiom, and in particular show that $\operatorname{Slice}$ is compatible only with some very weak form of $MA$.