On Thu, Aug 16, 2012 at 4:04 PM, meekerdb <meeke...@verizon.net> wrote:
> But there's also a different meaning of undecidable: a statement that can > be added as an axiom or it's negation can be added as an axiom > Axioms are important, you've got to be very careful with them! If you go around adding axioms at the drop of a hat it's a waste of time to prove anything because even if you are successful all you'll know is that there is a proof in a crappy logical system, you still will have no idea if it's true or not. For example, suppose you added the Goldbach Conjecture as a axiom and then a computer found a even integer greater than 4 that is not the sum of primes greater than 2, it would be a disaster, everything you've proved under that system would be nonsense. Axioms are supposed to be simple and self evidently true and Goldbach is not. > e.g. the continuum hypothesis within ZFC. > In 1940 Kurt Godel himself proved that if you add the continuum hypothesis to standard Zermelo-Fraenkel Set Theory you will get no contradictions. Then in 1962 Paul Cohen proved that if you add the NEGATION of the Continuum Hypothesis to standard set theory you won't get contradictions either. Together Godel and Cohen proved that the ability to come up with a proof of the Continuum Hypothesis depends on the version of set theory used. We were lucky with the Continuum Hypothesis, we know it's unprovable under Zermelo-Fraenkel so nobody spins their wheels trying to prove or disprove it, but not all unprovable statements are like that, Turing tells us that there are a infinite number of propositions that are unprovable that we can never know are unprovable. > Are there any explicitly known arithmetic propositions which must be true > or false under Peanao's axioms, but which are known to be unprovable? > I think you mean propositions about numbers that are true but cannot be shown to be true with Peano, if they are true or false under Peanao (and not true AND false!) then they are not unprovable. We know from Godel there must be a infinite number of such statements and we know from Turing there is no surefire way of detecting them all, and that's what makes them so dangerous, they are a endless time sink. And in fact although they are infinite in number as far as I know nobody has been able to point to a single one. So maybe trying to prove or disprove Goldbach is utterly pointless and maybe it is not, there is no way to know. John K Clark -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
