A mathematical incompleteness in Peano arithmetic

In Jon Barwise & H. Jerome Keisler (eds.), Handbook of Mathematical Logic. North-Holland Pub. Co.. pp. 90--1133 (1977)
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Slow Consistency.Sy-David Friedman, Michael Rathjen & Andreas Weiermann - 2013 - Annals of Pure and Applied Logic 164 (3):382-393.
Phase Transitions for Gödel Incompleteness.Andreas Weiermann - 2009 - Annals of Pure and Applied Logic 157 (2-3):281-296.
On the Number of Steps in Proofs.Jan Kraj\mIček - 1989 - Annals of Pure and Applied Logic 41 (2):153-178.
Informal Versus Formal Mathematics.Francisco Doria - 2007 - Synthese 154 (3):401-415.

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