$ \mathsf{TI}(\prec_{\varepsilon_0}) $ is transfinite induction on $\varepsilon_0$ ordinal notation by Cantor normal form. I think that proof-theoretic ordinal of $\mathsf{PA}+\mathsf{TI}(\prec_{\varepsilon_0})$ is more than $\varepsilon_0$, but I don't understand how to how to calculate it.
2026-03-25 01:28:03.1774402083
What is the proof-theoretic ordinal of $ \mathsf{PA}+\mathsf{TI}(\prec_{\varepsilon_0})$?
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