An algebraic characterization of power set in countable standard models of ZF

Extract

The following is a classical result:

Theorem 1.1. A complete atomic Boolean algebra is isomorphic to a power set algebra [2, p. 70].

One of the consequences of [3] is: If M is a countable standard model of ZF and is a countable (in M) model of a complete ℵ₀-categorical theory T, then there is a countable standard model N of ZF and a Λ ∈ N such that the Boolean algebra of definable (in T with parameters from ) subsets of is isomorphic to the power set algebra of Λ in N. In particular if and T the theory of equality with additional axioms asserting the existence of at least n distinct elements for each n < ω, then there is an N and Λ ∈ N with 〈P^N(Λ), ⊆〉 isomorphic to the countable, atomic, incomplete Boolean algebra of the finite and cofinite subsets of ω.

From the above we see that some incomplete Boolean algebras can be realized as power sets in standard models of ZF.

Definition 1.1. A countable Boolean algebra 〈B, ≤〉 is a pseudo-power set if there is a countable standard model of ZF, N and a set Λ ∈ N such that

It is clear that a pseudo-power set is atomic.