On Sun, Nov 9, 2014 at 6:02 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Goedel's original undecidable statement, as described in the article by Nagel & Newman about it in the 4-volume set The World of Mathematics, or in the more detailed book by Goedel's Proof by the same authors, is a statement that can be interpreted as saying "There is no proof of me."
So as you may have seen, if it's false then there *is* a proof of it, so it's true. So, if number theory is consistent, then it can't be false since that leads to a contradiction. So it must be true (in a cosmic sense).
I'm not sure what "true in a cosmic sense" means. The Goedel sentence for formal number theory is provable in ZF, so it's true if you believe in set theory. Also, the theory where you assume it's true is omega-consistent, while the theory where you assume it's false is omega-inconsistent, so that's a reason to prefer it (although I think proving this, or even stating it, requires set theory, rather than formal number theory). A theory is Omega-inconsistent if there is a formula P(X) such that all of P(0), P(1), P(2), P(3).... are provable, but so is "There exists an X such that not P(X)"
Is it plausible that every such undecidable proposition is that way because (maybe like the TWP) it is true -- or false -- only because of probabilistic accident?
Only if you can define "probabilistic accident" precisely. I don't think it's plausible that this can be done.
It seems to me that if we used a number system with infinitesimals (like the Surreals) to record probabilities, then most or all of the propositions like TWP that [appear to have probability 1 of being true] would have probability 1-eps of being true, where eps is the reciprocal of an uncountable number.
Two years ago, wouldn't you have put the proposition "there are infinitely many pairs of primes that differ by 1000 or less" to be in the same category of "statements with probability 1-eps of being true" as TWP? But it's not in any such mysterious category; it's just true. I see no reason not to think that TWP has the same status, and it will get proved in a year or a century.
If this is always the case, then the probability that ALL such propositions are true without exception is also 1. (For, the countable product of numbers of form 1-eps where eps is the reciprocal of an uncountable number would still be of the same form.)
Are there some propositions whose probability of being true at least heuristically can be calculated as a number strictly between 0 and 1 ???
The leap from "systems with infinitesimals exist" to "intuitive conclusions reached from intuitions about infinitesimals" is a large one. It's easy to reach contradictions by reasoning loosely about infinitesimals, so unless you define your terms very precisely, I find it hard to make any sense of statements like this. There's a heuristic principle that the primes behave like a randomly selected set of numbers, where the probability that n is selected is 1/ln(n), except when they don't. Formulating a precise version of this statement, proving such a statement, developing a theory of probability where probabilities are surreal numbers, and proving that the TWP has a meaningful probability in such a system, all seem to me much more difficult problems than TWP. Andy -- Andy.Latto@pobox.com