Abstract

We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF . It allows the use of set terms, but provides a static check of their validity. Like the inconsistent “ideal calculus” for set theory, it is essentially based on just two set-theoretical principles: extensionality and comprehension (to which we add ∈-induction and optionally the axiom of choice). Comprehension is formulated as: x ∈ {x | ϕ} ↔ ϕ, where {x | ϕ} is a legal set term of the theory. In order for {x | ϕ} to be legal, ϕ should be safe with respect to {x}, where safety is a relation between..