EDB — 26F

4.5 Z-F and Peano compatibility[26F]

Let’s go back now to the model ${\mathbb{N}}_{\text{ZF}}$ of $\mathbb{N}$ 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

It’s easy to see that N1 and N3 are true. The N5 property follows from the fact that ${\mathbb{N}}_{\text{ZF}}$ is the smallest set that is S-saturated. N2 and N4, derive from [1YM]↺↻.

We moreover saw in Theorem [24D]↺↻ that the relation $\subseteq$ satisfies the requisites of Hypothesis [26H]↺↻.