Re: approaching a proof


dave wrote:
> ken quirici wrote:
>
> > but suppose we have a theorem which we know
> > is unprovable in some system. Is it possible that there is an
> infinite
> > sequence of 'proofs' in that system that APPROACH a proof of the
> > theorem?
>
> Yes, in a slightly fuzzy sense, it can make sense to say
> that a sequence of proofs "approaches" a proof of a larger
> theorem. Here's an example.
>
> We don't know how to prove Goldbach's Conjecture, and for all
> we know, it may be unprovable in PA. However, we can argue
> heuristically that it is almost certainly true, using a density
> argument (we know the density of primes near N is 1/log(N), so
> we can use that to estimate the probability that a given even
> number is the sum of two primes). Then we can "prove" by
> calculation that every even number less than M is a sum of two
> primes, and then derive a "probability" p_M that all even numbers
> greater than M are a sum of two primes. Then we show that as M
> approaches infinity, p_M goes to zero,

I just want to correct a silly mistake. p_M goes to 1, not 0.

> so in a slightly fuzzy
> probabilistic sense, the limit of our proofs as M goes to infinity
> approaches a proof of Goldbach's Conjecture.

.

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