Abstract
Presumably, if the Gödelian is to find a solution to the Stability Problem for a given system T (T being an ideal system whose soundness is in question, and therefore a system whose syntax is to be represented or “arithmetized”) he must locate a set C of conditions on formulae of T (T now being treated also as the system in which the syntax of T is to be represented) such that (1) every formula of T that can reasonably be said to express the consistency of T satisfies the conditions in C, and (2) no formula of T that satisfies C can be proven in T provided that T is consistent. This being so, the Gödelian’s success in dealing with the Stability Problem will evidently depend crucially upon his ability to defend the reasonableness of his choice of C.
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Notes
The proof given here generally follows that of Smorynski [1977].
An example of this is the symmetrical version of the Rosser provability predicate studied in Kreisel and Takeuti [1974] (cf. pp. 15–16, 46–8.
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© 1986 Springer Science+Business Media Dordrecht
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Detlefsen, M. (1986). The Stability Problem. In: Hilbert’s Program. Synthese Library, vol 182. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7731-1_4
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DOI: https://doi.org/10.1007/978-94-015-7731-1_4
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