# Reference request - is there an axiomatic theory of consistency?

Is there an axiomatic theory whose domain of discourse can be interpreted as a collection of first order theories, which has a predicate $\mathrm{Con}$ such that $\mathrm{Con}(T)$ can be interpreted as saying that the theory $T$ is consistent?

More generally, is there an axiomatic theory of consistency?

Alternatively, is there an axiomatic theory of provability, with relation $\vdash$ such that $T \vdash t$ can be read `$T$ proves $t$.' That way, $\mathrm{Con}(T)$ can be defined as shorthand for $\neg(T \vdash \bot)$. In either case, I'd be interested in a reference recommendation.

EDIT: Yet another approach might be a theory with a (deductive) closure operator $D$ together with an extensional relation $\in$. Then $\bot \notin D(T)$ could be defined as saying $\mathrm{Con}(T)$.

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