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.