Journal
LOGIC JOURNAL OF THE IGPL
Volume 26, Issue 2, Pages 255-257Publisher
OXFORD UNIV PRESS
DOI: 10.1093/jigpal/jzx061
Keywords
Second incompleteness theorem; definable theories; Peano arithmetic; soundness
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Godel's second incompleteness theorem is generalized by showing that any Sigma(n+1)-definable and Sigma(n)-sound extension of Peano arithmetic (PA) cannot prove its own Sigma(n)-soundness. The optimality of the generalization is shown by presenting a Sigma(n+1)-definable and Sigma(n-1)-sound extension of PA that proves its own Sigma(n-1)-soundness.
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