I believe Goedel's original version of the theorem was formulated in terms
of systems of axioms containing the properties of the natural numbers, but
it can be extended to cover any system of axioms. I believe Penrose
discusses a proof based on Turing machines in his book "Shadows of the
mind" (which I _must_ finish someday :-().
At 5:44 PM 3/14/96 -0400, Steve Anonsen/GPS wrote:
>Allan Harvey wrote:
>[ secondary quote is from a mathematician friend of Allan's ]
>> >Suppose you have
>>>any collection of axioms that include the axioms that postulate the
>>>existence of the natural numbers. Then there are statements about the
>>>natural numbers that can be neither proved false nor true with these
>>>axioms.
>>
>>There would not seem to be any way to get from this the existence of things
>>you "know" to be true but can't prove. In fact, in the context of
>>mathematics and logic, the concept of "knowing" something to be true without
>>rigorously proving it seems entirely meaningless.
>>
>The key is that it can't be proven _with the axioms of that system_; if you
>are
>somehow outside that system, you may be able to decide the theorem (by having
>additional axioms, for example). But, there is no formal system with the
>axiomatic strength of the natural numbers that is complete in the sense that
>all theorems are decidable _within that system_ -- I believe that is Godel's
>theorem.
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