Re: Infinities

Dan Clemmensen (
Fri, 07 Nov 1997 19:51:52 -0500

Hal Finney wrote:
> The Continuum Hypothesis (CH) is known to be undecideable under current
> axioms of set theory.
[SNIP of really neat stuff]
> This is something they don't talk about much in math class - what do
> you do when your axioms have failed you and can't decide a question?
> What kind of reasoning works, how do you approach the problem? It is
> related to such issues as Penrose's "proof" that Godel's theorem implies
> that machines cannot think.
> The question is whether the task of finding new axioms is itself one
> which can be described formally and axiomatically, or equivalently,
> one which can be captured in a computer program. Or do we somehow
> have a non-axiomatic component which leads us to truth, as Penrose
> believes? This problem might be a good test case. It sounds like the
> preference is to set C = aleph-one, although maybe Godel was pushing
> for C = aleph-two. Would Penrose claim that we have an infallible
> non-axiomatic intuition which guarantees that we make the right choice?
> I don't understand his argument well enough to know what he would say.
> Hal

Wait a minute. I thought that Godel proved that any sufficiently
complex set of axioms (i.e., sufficient to use as a basis for
everyday mathematics) was provably incomplete or provably inconsistent.
A consistent system is one in which there is no statment which is
both provably true and provably false. A system that is complete
can prove that any statement in the system is either true or false.
It's trivial to prove that this implies that there are an infinite
number of undecidable statements in the system. Further, you can
choose to generate an infinite number of new consistent systems
by adding one of the undecidable statements as a new axiom. So,
if you like the Continuum Hypothesis, and it is provably undecidable
under the accepted axioms of set theory, you are free to create a
new consistent system by adding the CH as an axiom.

How am I supposed to go from this to Penrose's conclusion? Sounds
crazy to me.