We introduce the family of axioms, denoted $\operatorname{Slice}_κ$, that claim the existence of strictly increasing decompositions of the form $$2^δ=\bigcup_{α<κ} 2^δ\cap M_α,$$ where $δ<κ$, and $\{M_α|\; α<κ\}$ 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 $MA$.

v. 4.0, final version accepted to PAMS