Does a theory like Robinson's arithmetic have a proof-theoretic ordinal? If so, what is it?
2026-03-25 02:59:34.1774407574
proof theoretic ordinal for Robinson's arithmetic
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Well, since $Q$ doesn't even have induction along $\mathbb{N}$, I'm dubious that the notion of proof-theoretic ordinal makes sense for it; but if forced, I'd say the answer has to be $\omega$. Induction along finite orderings is trivial, so $\omega$ is the first ordinal for which it makes sense to ask "Does $Q$ prove induction along this ordinal?," and $Q$ doesn't.