This is a question asked out of curiosity, and because I can't understand the Wikipedia page.
I have often been told that PA cannot prove the validity of induction up to $\epsilon_0$, which has been expressed to me roughly as the claim that $\epsilon_0$ is well-ordered. I understand what ordinals are, and what $\epsilon_0$ is. I also understand first order logic and axiom schemes, so I understand how the induction axiom scheme formalizes the notion that $\omega$ is well-ordered.
What I don't understand is how one could formulate the statement that $\epsilon_0$ is well-ordered as a first order sentence in arithmetic. Would someone mind spelling this out for me?
Best Answer
Here's a more detailed answer:
The above-mentioned link constructs a recursive relation $E$ on $\omega$, such that $(\omega, E)$ is isomorphic to $(\epsilon_0, \in )$. Then, induction up to $\epsilon_0$ is interpreted as $E$-induction, that is, for every predicate $\phi$, if $(\forall x E y \phi(x))\rightarrow \phi(y)$ then $\forall y \phi(y)$.