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

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