Hilbert's Programme and Gödel's Theorems
Article first published online: 23 JUN 2005
Volume 56, Issue 4, pages 347–370, December 2002
How to Cite
Niebergall, K.-G. and Schirn, M. (2002), Hilbert's Programme and Gödel's Theorems. Dialectica, 56: 347–370. doi: 10.1111/j.1746-8361.2002.tb00250.x
- Issue published online: 23 JUN 2005
- Article first published online: 23 JUN 2005
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.