I want to coin a notion of "strong provability", to be defined as:


$S$ is strongly provable in $T$ if and only if there is a Gödel code of its proof in $T$ that is strictly smaller than any Gödel code of a proof of its negation in $T$


Formally:


$ S \text { is strongly provable in } T \iff \\ \exists x: \operatorname {Proof}\_T(x, \ulcorner S \urcorner ) \land \forall y (\operatorname {Proof}\_T (y, \operatorname {neg}(\ulcorner S \urcorner)) \implies x < y ) $


Now let $T \vdash S$ be the usual metatheoretic statement of $S$ being syntactically proved from $T$


Then is it provable that:


$(T\vdash S) \iff S \text { is strongly provable in } T$


The forward direction is clear, but the opposite is what irks me? Can we have a sentence whose smallest code of its proof (not the metatheoretic) is a non-standard natural, and every proof of its negations is strictly lager than it? I mean in this case it'll be indeterminate syntactically speaking, and the statements would be false, yet is this possible, can we have a theory that spells that and be consistent?



Can we have consistent theories stating opposing provability statements that are non-standardly coded?
This idea in play here is due to Rosser and is the main idea behind the [Gödel-Rosser theorem](https://en.wikipedia.org/wiki/Rosser%27s_trick).


Specifically, Rosser proposes to consider the sentence $\rho$ asserting that for every proof of $\rho$ in PA, there is a smaller proof of $\neg\rho$. In your terminology, $\rho$ asserts its own non-strong-provability. Such a sentence can be constructed by the fixed-point lemma. The conclusion is that if PA is consistent, then $\rho$ is independent of PA. It cannot be provable, since then it would have a proof of some specific length, and PA would prove that some smaller number would be a proof of $\neg\rho$, but by consistency none of those numbers can actually code a proof. And it cannot be refutable, since then $\neg\rho$ would have a proof of some specific length, and so PA would have to prove that one of the smaller numbers is a proof of $\rho$, which again can't happen by consistency.


The forward direction of your biconditional, which you say is "clear", is false if $T$ is inconsistent, since $T$ will prove every $S$, but it will not strongly prove every $S$. Meanwhile, the converse direction is true, since if a statement is strongly provable, it is provable.


Meanwhile, regarding the issues in your final paragraph, it may be interesting to consider the case of the Rosser sentence. In some models of PA, there is a proof of $\rho$ that is smaller than any proof of $\neg\rho$; and in other models of PA, there is a proof of $\neg\rho$ that is smaller than any proof of $\rho$. In both cases, those proofs are nonstandard, since $\rho$ is actually independent of PA.


