The Slicing Axioms

Abstract

We introduce the family of axioms, denoted ${\mathrm{Slice}}_{\kappa }$, 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 $\mathrm{Slice}$ is compatible only with some very weak form of $MA$.

arXiv commentv. 4.0, final version accepted to PAMS