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. [246]. 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
\[ S(x) {\stackrel{.}{=}}x∪ \{ x\} \quad . \]
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 [1YM].
We moreover saw in Theorem [24D] that the relation \(⊆\) satisfies the requisites of Hypothesis [26H].