Interesting function

For the discussion of math. Duh.

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oblivimous
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Interesting function

Postby oblivimous » Mon May 21, 2007 7:16 pm UTC

f(n) defined as the smallest number such that all proveable statements with a length of n characters can be proved in f(n) characters or less, and f(n) = 0 if there are no proveable statements with length n.

So if f(30) = 500 that means "Every 30-character-statement has a 500(or less)-character-proof or it cannot be proved at all"

I'm pretty sure this function exists for all whole number values of n. Although if we replace 'proveable' with 'true' then incompleteness may become an issue.

I think incompleteness also gives us this: there is no computable function g(n) such that we can know that g(n) > f(n) for all n

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Postby skeptical scientist » Mon May 21, 2007 9:13 pm UTC

In fact, there is no computable function g such that g(n)>f(n) for all n, even if we didn't know it (if there were, the set of provable statements would be computable, and that would give you a complete extension of PA).
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