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Godel's Incompleteness Theorems (view source)
Revision as of 19:08, June 13, 2008
, 19:08, June 13, 2008no edit summary
'''Gödel's Second Incompleteness Theorem''': If T is any consistent, sound, recursively enumerable first-order theory containing the axioms of Peano Arithmetic, then T cannot prove its own consistency. In particular, Peano Arithmetic cannot prove its own consistency. This means that the consistency of the axioms of Peano Arithmetic and stronger mathematical theories must be justified in other ways, usually by appealing to the fact that the axioms are obviously true or by appealing to the fact that no one has yet derived an inconsistency.
'''Gödel's Second Incompleteness Theorem''': If T is any consistent, sound, recursively enumerable first-order theory containing the axioms of Peano Arithmetic, then T cannot prove its own consistency. In particular, Peano Arithmetic cannot prove its own consistency. This means that the consistency of the axioms of Peano Arithmetic and stronger mathematical theories must be justified in other ways, usually by appealing to the fact that the axioms are obviously true or by appealing to the fact that no one has yet derived an inconsistency.
==Incompleteness and God==
Incompleteness is sometimes used to refute the existence of an [[omnipotent]] god: If [[God]] were omnipotent, he would act as an [[oracle]] (in the mathematical sense) for any mathematical theory. That is, he could decide for any statement S whether S or not-S were true, which contradicts incompleteness. This argument is a more [[formal]] version of the "Can God make a rock even He could not move" [[paradox]].
[[Category:Logic]][[Category:Mathematics]]
[[Category:Logic]][[Category:Mathematics]]