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?

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 $\varepsilon_0$, which has been expressed to me roughly as the claim that $\varepsilon_0$ is well-ordered. I understand what ordinals are, and what $\varepsilon_0$ 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 $\varepsilon_0$ is well-ordered as a first order sentence in arithmetic. Would someone mind spelling this out for me?

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?

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?

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?

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?