standard model of $\mathbb{N}$ and true $\Pi^0_1$ sentences

Question

Does the fact that provability of some true $\Pi^0_1$ sentences is equivalent to the existence of particular (I do not know from the top of my head to which ones, does someone know how they are called) large cardinals really speak in favor to their existence ?

Answer 1

This question is based on a (common) misunderstanding. No arithmetic (let alone $\Pi^0_1$) sentence can imply the existence of any large cardinal. There is a simple reason for this, which I'll illustrate in the specific case of (strongly) inaccessible cardinals.

Suppose $\alpha$ is a consistent-with-$\mathsf{ZFC}$ arithmetic sentence. (This is a bit of an abuse of notation - really, $\alpha$ is the set-theoretic encoding of a statement of the form "$(\mathbb{N};+,\times)\models\varphi$" for some $\varphi$ - but meh.) Let $M\models\mathsf{ZFC+\alpha}$. If $M$ has no inaccessible cardinals, we're done. Otherwise, let $\kappa$ be (the thing $M$ thinks is) the least inaccessible. The substructure $$N=(V_\kappa)^M,$$ basically "$M$ cut off at height $\kappa$," is again a model of $\mathsf{ZFC}$. Moreover, since $\kappa$ was the least inaccessible in $M$, we have $N\models$ "There are no inaccessible cardinals." Finally, since $N$ and $M$ have the same $\omega$, they agree on arithmetic sentences so $N\models\alpha$.

This general strategy works (occasional with mild tweaks, e.g. for weakly inaccessible cardinals we need to use $(L_\kappa)^M$ instead of $(V_\kappa)^M$) for any sort of large cardinal principle.

What is true is that $\Pi^0_1$ sentences can imply the consistency of large cardinal hypotheses. There's no surprise here, though, since all consistency statements are $\Pi^0_1$ properties, so large cardinals aren't playing any special role here. In particular, $\Pi^0_1$-ness of consistency can't possibly be construed as evidence for consistency, let alone truth.