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Paper IPM / M / 14498  


Abstract:  
GÃ¶delâRosser's Incompleteness Theorem is generalized by showing Î n+1incompleteness of any Î£n+1definable extension of Peano Arithmetic which is either Î£nsound or nconsistent. The optimality of this result is proved by presenting a complete, Î£n+1definable, Î£nâ1sound, and (nâ1)consistent theory for any n>0â . Though the proof of the incompleteness theorem for Î£n+1definable theories using the Î£nsoundness assumption is constructive, it is shown that there is no constructive proof for the Incompleteness Theorem for Î£n+1definable theories using the nconsistency assumption, when n>2â .
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