At 03:02 PM 11/9/2014, Dan Asimov 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). Therefore there really is no proof of it.
QUESTIONS: ----------
Are there relatively simple propositions in number theory that are known to be undecidable?
Hilbert's Tenth Problem is about as simple as you can get.
What about undecidable in the sense that we have no way of knowing whether the proposition is true or false. (Potentially, like the Twin Prime Conjecture.)
Undecidable in this sense usually means that the new statement can be added as a new axiom without rendering the system inconsistent. Alternatively, the negative of the new statement can be added as a new axiom without rendering the system inconsistent. So you get two new non-trivial theories: one in which the previously undecidable statement is true, and one in which the previously undecidable statement is false.
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?
???
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.
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 ???
There has been a huge amount of work on "probabilistic logics", both classical & quantum. There are also so-called "modal logics" which define quantifiers "necessary" and "possible", and there is some intersection with probabilistic logics & quantum logics. There may be some versions of these logics that deal with uncountable universes, but I'd have to do some searching to find out. Countably infinite universes -- e.g., the integers -- have also been studied to some extent via theories such as "S1S", "S2", "real closed fields", "Presburger Arithmetic", etc.