4.5 Z-F and Peano compatibility[26F]

Let’s go back now to the model \(ℕ_\text {ZF}\) of \(ℕ\) built relying on the theory of Zermelo—Fraenkel, seen in Sec. 3.8. We want to see that this model satisfies Peano’s axioms.

Recall that, given \(x\) (any set, not necessarily natural number) the successor is defined as

It’s easy to see that N1 and N3 are true. The N5 property follows from the fact that \(ℕ_\text {ZF}\) is the smallest set that is S-saturated. N2 and N4, derive from 5.

We moreover saw in Theorem 91 that the relation \(⊆\) satisfies the requisites of Hypothesis 143.