1. primality test: for any arbitrary P fixed in advance, for any epsilon, you can verify P’s primality with confidence (1-epsilon) by doing 1-epsilon. But, unlike primality tests, this procedure tells you nothing about the complexity of an R fixed in advance.

Let N be a nice big number, like Ackermann(10000000000) or something like that. Suppose we can experimentally verify Goldbach’s conjecture for all even numbers up to N. We might use our experiment to form a Bayesian prior hypothesis about numbers larger than N, i.e. “for k>N, the probability that k is a counterexample to Goldbach’s conjecture is at most 1/N”. But, big as N is by the standards of normal human experience, it’s smaller than almost all of the finite natural numbers. We really can’t infer anything about “for all k” from that hypothesis. There is no probability distribution over the entire set of naturals that we can claim to have sampled.

The coin flipping experiment, on the other hand, samples from a known, finite distribution. The sample space is the 2**(2e6) strings of 2 million bits. The number of such strings with Kolmogorov complexity <= 1 million is approx 2**(1e6) which means the probability of getting such a string from random flips is approx 2**(-1e6), and the probability that the assertion is true is about (1-2**(-1e6)). Of course you can make it arbitrary close to 1 by flipping more times. We really do get a quantifiable probability bound on the statement, just like with a pseudoprime test.

There is a rather powerful implicit axiom here, namely that flipping coins actually gives Kolmogorov-random bits. They may not be perfectly p=0.5 iid since they are real coins which might have a tiny bias or correlation, but we can make fairly good estimates of the Shannon entropy which should be close to 1 bit per flip. So, our creaky old physical universe is somehow capable of accessing mathematical truth (abductive inference) that is beyond being deducible from axioms.

What’s more, if we subscribe to current beliefs about complexity theory, we really do need testimonial evidence (someone saying they actually saw the coins being flipped) to accept the statement. The stronger forms of P/=NP say that we can make a cryptographic pseudorandom number generator that, given (say) a 128-bit uniform-random seed, can expand it to arbitrary size pseudorandom strings that can’t be distinguished from true random bits by any feasible (i.e. P-time) computational process (of course it’s trivial in exp-time).

Consider the idea of a normal number (that is, one such that in every base, all digits have equal limiting frequency in the representation of the number). It is not hard to prove that in the interval [0,1], the set of normal numbers has measure 1. However, no one thinks that this gives anything like a probabilistic proof that 1/pi or 1/e is normal. It’s not clear to me how your example differs significantly from this, except that there is a positive probability of error rather than probability 0 of error.

The Chaitin-type statements are interesting ones to consider though, since there is some very natural sense in which they are “very likely” to be true, despite the fact that they are unprovable.

(This example inspired by Leonid Levin’s article here: http://www.cs.bu.edu/fac/lnd/expo/gdl.htm see his example of rolling dice to make an incompressible string)