# A theory which seems to have proof-theoretic ordinal $\omega_1^{CK}$

I'm trying to understand proof-theoretic ordinals, and mistakenly "proved" there's a sound recursive theory of arithmetic with proof-theoretic ordinal $$\omega_1^{CK}.$$ That's impossible, so where does this go wrong?

"Proof:" Fix a recursive ordering $$<_H$$ on $$\omega$$ of length $$\omega_1^{CK}(1+\eta),$$ $$\eta$$ the order type of the rationals, with each nonempty hyperarithmetic set having a $$<_H$$-minimal element. Let $$<_n$$ be the initial segment of $$<_H$$ below $$n.$$ Then $$<_n$$ is a recursive ordering. Call $$n$$ $$\textit{good}$$ if the theory $$S_n:=Z_2+<_n$$ is well-founded$$"$$ is arithmetically sound.

The set of good $$n$$ is hyperarithmetic and contains the well-founded part of $$<_H,$$ so there is some $$n^*$$ which is good and lies in the ill-founded part of $$<_H.$$ Then $$T=$$ PA $$+S_{n^*}$$ is arithmetically sound$$"$$ is sound and recursive. Notice that for each $$\alpha<\omega_1^{CK},$$ there is $$n$$ such that $$<_n$$ is isomorphic to $$\alpha,$$ and $$T$$ proves $$<_n$$-induction. So $$T$$ is as desired.

Having read this post from Dmytro (https://mathoverflow.net/a/278615/109573), I think the answer here is that proof-theoretic ordinals are best defined for $$\Pi_1^1$$-sound theories which interpret a weak second order arithmetic (e.g. ACA$$_0$$). While we can reasonably assign weak first order theories (e.g. subsystems of PA) proof-theoretic ordinals based on how much induction they prove, this quickly becomes problematic.
For example, one might want to assign PA $$+ \epsilon_0$$-induction a proof-theoretic ordinal greater than $$\epsilon_0.$$ But then we would expect the second-order theory ACA$$_0$$ + arithmetic $$\epsilon_0$$-induction to have at least as high of a proof-theoretic ordinal, even though, if I'm not mistaken, this theory does not prove the well-foundedness of $$\epsilon_0$$ (at the very least, it's clear ACA$$_0$$ + arithmetic $$\alpha$$-induction does not prove well-foundedness of $$\alpha$$ for arbitrary recursive ordinals $$\alpha$$). So strong theories of first-order arithmetic probably don't have a clear proof-theoretic ordinal.