# How do we know some ordinal $n$ is the proof-theoretic ordinal of some theory?

This might be an elementary question, and I apologize for that.

I've been dabbling in ordinal analysis / proof theory for a few days, back when I read W. Buchholz, “A new system of proof-theoretic ordinal functions”. Then, I read up on ordinal analysis itself on Wikipedia. There, it states that:

1. The proof-theoretic ordinal $$n$$ of some theory $$\mathsf{T}$$ is the supremum of all ordinals the theory can prove is founded.
2. The proof-theoretic ordinal for Robinson arithmetic, $$\mathsf{Q}$$, is $$\omega$$ (and I'm assuming $$\omega$$ is defined as $$\{0,1,2,\dots\}$$.

So my question is:

How do we know that $$\omega$$ is the proof-theoretic ordinal? Couldn't it be some other ordinal $$n$$ where $$n>\omega$$? Is there some method of proving the ordinal is infact the proof-theoretic ordinal?

E: It should hereby be noted that the example of $$\mathsf{Q}$$, mentioned here, is just an example -- the question applies to other theories, and is a "general" question; i.e. how does this work for other theories?

The question mentioned does help explain that $$\omega$$ is in fact the proof-theoretic ordinal for $$\mathsf{Q}$$, but that it does not elaborate on how one knows that $$\omega$$ is the supremum of $$\mathsf{Q}$$, as well as elaborating on the method of finding a supremum for any given theory $$\mathsf{T}$$, as I elaborated above.

• As wiki says $\omega$ is the set of all natural numbers, so what should $n\ge \omega$ mean? Commented Aug 11 at 19:32
• @trula: $n$ is an ordinal here, not a natural number. Commented Aug 11 at 20:15
• In order for it to be $>\omega$, $Q$ would have to prove induction on $\omega$. (Though as Wikipedia alludes to, Q is so weak that concepts such as this struggle to make sense.) Commented Aug 12 at 0:39
• This question is similar to: proof theoretic ordinal for Robinson's arithmetic. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. Commented Aug 12 at 1:46