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Apparently, see Feferman or Wikipedia, in a consistent system there are formulations of consistency that are demonstrable in the system itself while others are not. What distinguishes one from another? For example, isn't Gödel's second incompleteness theorem valid for any formulation of consistency of P? If not, why is his proof valid for his formulation of consistency and not for others?

Speltzu
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    The second incompleteness theorem holds (among others) for provability predicates satisfying the Hilbert–Bernays–Löb derivability conditions. You can find that on Wikipedia. This isn’t really a question suitable for this site; it might be more appropriate at https://math.stackexchange.com . – Emil Jeřábek Jan 25 '24 at 20:22
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    @EmilJeřábek, I think that this question would not necessarily have much luck getting a good answer on MSE, so really is fine here. One of my criteria is whether, concerning things that I've had some interest in over the years, I can or cannot immediately see/know "the answer". Here, I could not, so I respect this as a reasonable question. :) – paul garrett Jan 25 '24 at 20:51
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    I also think that MSE would be more appropriate for this question. – Marco Ripà Jan 26 '24 at 00:53
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    As I already mentioned, the needed derivability conditions are explained in the Wikipedia article on Gödel's incompleteness theorems (and they have a more detailed dedicated article of their own). That, for me, is a clear indication that the question is not research-level. Furthermore, the OP should already know this, considering that he refered to the Wikipedia article in the question itself, so he just needs to read more carefully. – Emil Jeřábek Jan 26 '24 at 07:06
  • The problem is then to construct a provability predicate that satisfies the Hilbert–Bernays provability conditions, that might not even exist. – Speltzu Jan 26 '24 at 11:07
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    Similar questions have been asked and answered before on this site; e.g., Clarification of Gödel's second incompleteness theorem, Is there a consistent arithmetically definable extension of PA that proves its own consistency?, In what sense does the sentence con(PA) "say" that PA is consistent? See also Relationship between first and second incompleteness theorems. – Timothy Chow Jan 26 '24 at 12:16
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    To answer the question in a nutshell, one can write down provability predicates Prv($\varphi$) that are extensionally correct but intensionally incorrect; i.e., they don't correctly express or say that "$\varphi$ is provable," even though Prv($\varphi$) happens to be true if and only if $\varphi$ is provable. Gödel's second incompleteness theorem does not apply to these intensionally incorrect predicates. – Timothy Chow Jan 26 '24 at 21:37
  • @Timothy Chow That's like saying that Gödel's second incompleteness theorem is true because it only applies to those consistency formulations for which Gödel's second incompleteness theorem is true. – Speltzu Jan 27 '24 at 07:55
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    @Speltzu If you actually want to learn something about this subject, you should look up what "extensional" and "intensional" mean, rather than try to pick a fight in the comments while speaking out of ignorance. – Timothy Chow Jan 27 '24 at 11:52
  • It seems like as if we know that Gödel's second incompleteness theorem is false but we refuse to admit it, and so we construct tailor-made consistency formulations so that it is not. – Speltzu Jan 27 '24 at 15:37

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