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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?

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    $\begingroup$ 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 math.stackexchange.com . $\endgroup$
    – Emil Jeřábek
    Commented Jan 25 at 20:22
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    $\begingroup$ @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. :) $\endgroup$
    – paul garrett
    Commented Jan 25 at 20:51
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    $\begingroup$ 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. $\endgroup$
    – Timothy Chow
    Commented Jan 26 at 12:16
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    $\begingroup$ 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. $\endgroup$
    – Timothy Chow
    Commented Jan 26 at 21:37
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    $\begingroup$ @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. $\endgroup$
    – Timothy Chow
    Commented Jan 27 at 11:52

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