Can Modalities Save Naive Set Theory?

History The late Grisha Mints asked Scott whether a naive set theory could be consistent in modal logic. Here are two modal forms of comprehension: (∃y)(∀x)(x ∈ y ↔ ϕ) (Comp) (∃y)(∀x)(x ∈ y ↔ ϕ) (Comp) At the time (2009) neither he nor Scott knew the answer. In the most commonly used systems, where the Converse Barcan Formula (∀xϕ → ∀xϕ) is derivable, (Comp) follows from another comprehension principle: (∃y)(∀x)(x ∈ y ↔ ϕ) In lectures Scott had presented a modal version of ZF which uses: (∃y)(∀x)(x ∈ y ↔ x ∈ u ∧ ϕ) (MZF Comp)