Hilbert’s Program pp 93-141 | Cite as

# The Stability Problem

## 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.

## Keywords

Stability Problem Mathematical Proof Mathematical Practice Derivability Condition Logical Technique## Preview

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## Notes

- 1.The proof given here generally follows that of Smorynski [1977].Google Scholar
- 19.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.Google Scholar