All Questions

As I currently understand it, induction on formulas containing $N+1$ first-order quantifiers is required to prove the well-ordering of the ordinal $(\omega \uparrow\uparrow N) < \epsilon_0$, that ...

Consider the Peano axioms. There exists a model for them (namely, the natural numbers with a ordering relation $<$, binary function $+$, and constant term $0$). Therefore, by the model existence ...