The following is from page $19$ of Holz' Introduction to Cardinal Arithmetic.
As I understand it (see here), ordinary and complete induction (on $\omega$) work as follows:
- Ordinary Induction: if $P(0)$ holds and if $P(k)\implies P(k+1)$ for any $k$, then $P(n)$ holds for any $n$.
- Complete Induction: if $P(0)$ holds and if $P(1)\land\cdots\land P(k)\implies P(k+1)$ for any $k$, then $P(n)$ holds for any $n$.
Shouldn't Corollary 1.2.8 be called ordinary induction? As I see it, complete induction would instead take a form such as $$A\subseteq \omega\land 0\in A\land\forall \alpha\in A (\beta <\alpha \implies \beta\in A)\implies A=\omega.$$