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Recursively enumerable (r.e.) theories are theories that can be written down in a mechanical way.  The restriction to recursively enumerable theories is required, since it is easy to construct non-r.e. theories that are complete--for example, the theory that contains exactly all true statements of number theory is complete, but cannot be written down in a mechanical way.

Recursively enumerable (r.e.) theories are theories that can be written down in a mechanical way.  The restriction to recursively enumerable theories is required, since it is easy to construct non-r.e. theories that are complete--for example, the theory that contains exactly all true statements of number theory is complete, but cannot be written down in a mechanical way.

The first incompleteness theorem is itself a theorem of Peano Arithemetic.  A corollary of this fact, noted independently by [[John von Neumann]] and Godel, is the:

The first incompleteness theorem is itself a theorem of Peano Arithmetic.  A corollary of this fact, noted independently by [[John von Neumann]] and Godel, is the:

'''Godel'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.

'''Godel'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.

[[category:mathematics]]

[[category:mathematics]]