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| Paper IPM / M / 14498 |
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| Abstract: | |
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GödelâRosser's Incompleteness Theorem is generalized by showing Î n+1-incompleteness of any Σn+1-definable extension of Peano Arithmetic which is either Σn-sound or n-consistent. The optimality of this result is proved by presenting a complete, Σn+1-definable, Σnâ1-sound, and (nâ1)-consistent theory for any n>0â . Though the proof of the incompleteness theorem for Σn+1-definable theories using the Σn-soundness assumption is constructive, it is shown that there is no constructive proof for the Incompleteness Theorem for Σn+1-definable theories using the n-consistency assumption, when n>2â .
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