  We introduce the family of axioms $\operatorname{Slice}_\kappa$, which claim
existence of non-trivial decompositions of the form $\{2^{<\kappa} \cap
V_\alpha |\; \alpha<\kappa\}$, for sets $2^{<\kappa}$, where $\{V_\alpha|\;
\alpha<\kappa \}$ is a sequence of transitive models of set theory. We study
compatibility of these axioms with versions of Martin's Axiom, and in
particular show that $\operatorname{Slice}_{\omega_1}$ is consistent only with
some very weak form of $MA$.
