Sentence examples for predicate provability from inspiring English sources

The answer is unfortunately a resounding no: Vardanyan (1986) has proved on the basis of ideas by Artemov (1985) that the set of sentences of predicate provability logic all of whose realizations are provable in PA is not even recursively enumerable, so it has no reasonable axiomatization.

Finally, one can of course study predicate provability logic.

Is there a nicely axiomatized predicate provability logic that is adequate, proving exactly the valid principles of provability?

Roughly put, (D2) requires that the whole demonstration of (D1), for the candidate provability predicate ProvF, can itself be formalized inside F. Finally, (D3) requires that the provability predicate is closed under Modus Ponens.

These rely on different formalizations of the provability predicate PrT than the standard ones.

It is always possible to choose the provability predicate ProvF x) to be a Σ01-formula.

For example, Rosser's provability predicate mentioned above would not do; one can prove the "consistency" of F in F, if consistency is expressed in terms of Rosser's provability predicate.

It is known that formalized consistency statements are unprovable whenever the provability predicate obeys certain general derivability conditions.

This theorem says essentially that the modal logic GL captures everything that Peano Arithmetic can truthfully say in modal terms about its own provability predicate.

The basic reason for this is that, unlike in the first theorem, not just any, merely extensionally adequate provability predicate works for the formalization of the consistency claim.

The proof of the second incompleteness theorem requires that the provability predicate in F satisfies a number of conditions which are used in the details of the proof.