GENERAL NOTIONS OF CONSISTENCY

Abstract We explore general notions of consistency . These notions are sentences $\mathcal {C}_{\alpha }$ (they depend on numerations $\alpha $ of a certain theory) that generalize the usual features of consistency statements. The following forms of consistency fit the definition of general notions of consistency ( ${\texttt {Pr}}_{\alpha }$ denotes the provability predicate for the numeration $\alpha $ ): $\neg {\texttt {Pr}}_{\alpha }(\ulcorner \perp \urcorner )$ , $\omega \text {-}{\texttt {Con}}_{\alpha }$ (the formalized $\omega $ -consistency), $\neg {\texttt {Pr}}_{\alpha }(\ulcorner {\texttt {Pr}}_{\alpha }(\ulcorner \cdots {\texttt {Pr}}_{\alpha }(\ulcorner \perp \urcorner )\cdots \urcorner )\urcorner )$ , and $n\text {-}{\texttt {Con}}_{\alpha }$ (the formalized n -consistency of Kreisel). We generalize the former notions of consistency while maintaining two important features, to wit: Gödel’s Second Incompleteness Theorem, i.e., (with $\xi $ some standard $\Delta _0(T)$ -numeration of the axioms of T ), and a result by Feferman that guarantees the existence of a numeration $\tau $ such that $T\vdash \mathcal {C}_\tau $ . We encompass slow consistency into our framework. To show how transversal and natural our approach is, we create a notion of provability from a given $\mathcal {C}_{\alpha }$ , we call it $\mathcal {P}_{\mathcal {C}_{\alpha }}$ , and we present sufficient conditions on $\mathcal {C}_{\alpha }$ for the notion $\mathcal {P}_{\mathcal {C}_{\alpha }}$ to satisfy the standard derivability conditions. Moreover, we also develop a notion of interpretability from a given $\mathcal {C}_{\alpha }$ , we call it $\rhd _{\mathcal {C}_{\alpha }}$ , and we study some of its properties. All these new notions—of provability and interpretability—serve primarily to emphasize the naturalness of our notions, not necessarily to give insights on these topics.