VARIANTS OF KREISEL’S CONJECTURE ON A NEW NOTION OF PROVABILITY

Abstract Kreisel’s conjecture is the statement: if, for all $n\in \mathbb {N}$ , $\mathop {\text {PA}} \nolimits \vdash _{k \text { steps}} \varphi (\overline {n})$ then \forall x.\varphi (x)$ . For a theory of arithmetic T given recursive function h $T _{\leq h} $ holds if there proof $\varphi in whose code at most $h(\#\varphi )$ This notion depends on underlying coding. ${P}^h_T(x)$ predicate $\vdash h}$ It shown that exist sentence and total such $T\vdash h}\mathop {Pr}} _T(\ulcorner \mathop \urcorner )\rightarrow but where _T$ stands standard provability statement related to by Montagna. Also variants weakenings are studied. By use reflexion principles, one can obtain $T^h_\Gamma extends version holds: $\Gamma -formula (where an arbitrarily fixed class formulas) that, Derivability conditions studied satisfy following implication: corresponds arithmetization conjecture. certain theories, exists steps}}\ \subseteq\