@DominicvanderZypen I see your point. Thanks.

I understand that this might not be precisely in the flavor that OP was asking for, but I'd gladly hear what the person who down-voted has to say.

What are some references for this result?

No, sorry - I know this just as a "well-known result", but I don't know a reference.

Arithmetical statements can be a way to go - iirc, forcing cannot change truth of any arithmetic statement.

Do you know of any attempt to explicitly write down a $\Delta_0$ formula which defines factorial? Following the steps of the proof you sketch would probably lead to quite long formula, but there might be some way to give it shorter.

I'd say it's all about simulation. Check out some cellular automaton like Golly. If you want to check out some mechanisms, this is a good place to start conwaylife.com/wiki/Main_Page

My guess is that there is no real notion of "strongness", and "stronger" used in the above example is just a common terminology (without any real basis to call it so).

I believe in set-theoretic considerations one would just write $\Pi_3, \Delta_0$, because iirc superscript is used when writing (higher-order) arithmetical statements.

I'm quite certain that sentence in question implies consistency of PA.

Turing machines and related usually cannot work with general ordinals, because they are infinitary concepts. Could you specify what you mean with "computable" in this case?

How does one define arithmetical set in the context of sets of sets?