A formal reduction of the general problem of the expressibility of formulas in the G\"odel-L\"ob provability logic

M. F. Raţă

Diskretnaya Matematika, Tome 14 (2002) no. 2, pp. 95-106

Résumé

It is a well-known idea to embed the intuitionistic logic into the modal logic in order to interprete the modality ‘provable’ as the deducibility in the Peano arithmetics with also well-known difficulties. P. M. Solovay and A. V. Kuznetsov introduced into consideration the Gödel–Löb provability logic whose formulas are constructed from propositional variables with the use of the connectives $\$, $\vee$, $\supset$, $\neg$, and $\Delta$ (Gödelised provability). This logic is defined by the classic propositional calculus complemented by the three $\Delta$-axioms $$ \Delta(p\supset q)\supset(\Delta p\supset\Delta q),\quad \Delta(\Delta p\supset p)\supset\Delta p,\quad \Delta p\supset\Delta\Delta p, $$ and the extension rule (the Gödel rule). A formula $F$ is called (functionally) expressible in terms of a system of formulas $\Sigma$ in logic $L$ if, on the base of $\Sigma$ and variables, it is possible to obtain $F$ with the use of the weakened substitution rule and the rule of change by equivalent in $L$. The general problem of expressibility in a logic $L$ requires to give an algorithm which for any formula $F$ and any finite system of formulas $\Sigma$ recognises whether $F$ is expressible in terms of $\Sigma$ in $L$. In this paper, it is proved that in the Gödel–Löb provability logic and in many of extensions of this logic there is no algorithm which recognises the expressibility. In other words, the general expressibility problem is algorithmically undecidable in these logics.