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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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1 Answer 1

up vote 2 down vote accepted

There is a field called Provability Logic. Its basic (and difficult) result is that, in a suitable logic, Loeb's theorem captures all the reasoning that Peano arithmetic can perform about provability.

http://plato.stanford.edu/entries/logic-provability/#2

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Not quite what I had in mind, but it's an awesome starting point nonetheless. –  goblin Feb 4 '13 at 19:55
    
There are some proof theorists and logicians who post on MSE, if you unaccept this answer they might be likelier to see the question. I'm just giving the well known superficial answer. –  zyx Feb 5 '13 at 4:17
    
Thanks for the tip! –  goblin Feb 5 '13 at 4:27

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