Some consequences of the axiom of power-set

Extract

In this paper we prove:

Theorem 1. Any finite model of the axiom of power-set also satisfies the axioms of extensionality, sum-set and choice.

Clearly, it will follow from (2) below that in a finite model the axiom of power-set is satisfied if and only if every set is a power-set. Thus, Theorem 1 follows immediately from Theorem 2 below, where by a theory of sets we mean a first-order theory without identity and with only one binary predicate symbol ∈.

Theorem 2. If in a theory of sets every set is a power-set and if the axiom of power-set is valid, then the axioms of extensionality, sum-set and choice are valid.

The proof of Theorem 2 will follow from the lemmas which we establish below.

We mean by x = y that x and y have the same elements. We denote a power-set of x by P(x) when it exists; similarly, we denote a sum-set of x by Ux.

Clearly, in every theory of sets we have:

(1) (x ⊂ y) ↔ (P(x) ⊂ P(y)),

(2) (x = y) ↔ (P(x) = P(y)),

(3) (x = y) → ((x ∈ P(z)) → (y ∈ P(z))),

(4) ⋃P(x) = x.

In view of (2), (3) and the definition of equality, we have:

Lemma 1. If in a theory of sets every set is a power-set, then equal sets are elements of the same sets.

We have also, in view of (4):

Lemma 2. If in a theory of sets every set is a power-set, then every set has a sum-set.