Simply, if we define a binary property ‘P’ that can only be tested by algorithms that might not halt, and show, say by computation, that every natural number up to some arbitrarily large number ‘N’ has the property P, does that mean that there must be a generalized deductive proof that for all natural numbers P holds?
Yes...
How big can N get before there must be a deductive proof that P holds for all natural numbers?
Unfortunately, very big, as a rule, and not explicitly computable.
What if N was larger than Graham’s number[1]?
Graham’s number would not be nearly large enough for “interesting” conjectures, “interesting” being more complex (by Kologomorov’s definition) than the specification of Graham’s number.
Well, in such a case you can use the method to test somewhat more complex conjectures.
Unfortunately, the growth of the Busy Beaver function is so rapid that the number of bits of complexity of such provable conjectures added by your revised definition of N (I am ignoring the word “larger”, since that makes N undefined), is likely not very many.
Yes...
Unfortunately, very big, as a rule, and not explicitly computable.
Graham’s number would not be nearly large enough for “interesting” conjectures, “interesting” being more complex (by Kologomorov’s definition) than the specification of Graham’s number.
See http://en.wikipedia.org/wiki/Busy_beaver and the material referenced therein.
What if N is larger than the number you get when you plug graham’s as both of the inputs to the ackerman function?
Well, in such a case you can use the method to test somewhat more complex conjectures.
Unfortunately, the growth of the Busy Beaver function is so rapid that the number of bits of complexity of such provable conjectures added by your revised definition of N (I am ignoring the word “larger”, since that makes N undefined), is likely not very many.