Remark
186
[26K]Peano’s Axioms are provable in this model; moreover the order relation satisfies the requirements of Hypothesis [26H]; therefore this model of \(ℕ\) enjoys all properties discussed in Sec. [1X9]: all different versions of the induction principle; \((ℕ,⊆)\) is well-ordered; definitions by recursion; arithmetic, etc.
When we will want to compare this model with other models, we will denote it by \(ℕ_\text {ZF}\).