Proof of Induction on ℕ₀ from 0 for a Condition Expressed by a Predicative Formula

Proof of (i). Let $\psi$ be a predicative formula expressing $\varphi$, as in Conditions Expressed by a Predicative Formula, and Guards (ii): its free variables are among $x$ and the parameters of $\varphi$, and, the parameters having their fixed values, for every set $a$ the formula $\psi$ holds with $a$ assigned to $x$ if and only if $\varphi(a)$. Taking $\mathbb{N}_{0}$ as an additional parameter, let $Z=\{x\in\mathbb{N}_{0}:\psi\}$ be the class of Class Abstraction: the Class of All Sets Satisfying a Predicative Formula (ii). By Class Abstraction: the Class of All Sets Satisfying a Predicative Formula (i), for every set $a$,

$$a\in Z\quad\text{if and only if}\quad a\in\mathbb{N}_{0}\ \text{and}\ \varphi(a).$$

We have $0\in\mathbb{N}_{0}$, and $\varphi(0)$ holds by hypothesis, so $0\in Z$. Let $k\in\mathbb{N}_{0}$ with $k\in Z$. Then $\varphi(k)$, so $\varphi(S(k))$ by hypothesis; and $S(k)\in\mathbb{N}_{0}$ by Omega Is the Least Inductive Class: It Is a Set, Induction from Zero, the Peano Properties, and Transitivity (ii). Hence $S(k)\in Z$. By Omega Is the Least Inductive Class: It Is a Set, Induction from Zero, the Peano Properties, and Transitivity (iv), $\mathbb{N}_{0}\subseteq Z$, so every $k\in\mathbb{N}_{0}$ lies in $Z$ and $\varphi(k)$ holds.

Proof of (ii). For every $k\in\mathbb{N}_{0}$ the successor of $k$ is $S(k)=k+1$, by Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication (v). So the hypotheses say that $\varphi(0)$ holds and that $\varphi(k)$ implies $\varphi(S(k))$ for every $k\in\mathbb{N}_{0}$, and (i) gives $\varphi(n)$ for every $n\in\mathbb{N}_{0}$.