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Abstract

In this paper, we attempt to show that a weak version of Hilbert's metamathematics is compatible with Gödel's Incompleteness Theorems by employing only what are clearly natural prov- ability predicates. Defining first “T proves the consistency of a theory S indirectly in one step”, we subsequently prove (i) “PA proves its own consistency indirectly in one step” and sketch the proof for (ii) “If S is a recursively enumerable extension of (QF-IA), S proves its own consistency indirectly in one step”. The formalizations of the metatheoretical consistency assertions that occur in these theorems are clearly the natural ones. We conclude the paper with reflections on indirect consistency proofs and soundness proofs.