Re: CHAT: mathematics

On Sat, 18 Nov 2000, Dennis Paul Himes wrote:

>     Goedel proved that a consistent theory's G is neither provable nor
> refutable in that theory (and therefore, by Goedel's Completeness Theorem,
> neither true nor false).
Right, but that says nothing about the provability of ~G.

> If G is refutable in, say, Peano Arithmetic, then
> PA would be inconsistent, which is a possibility, but one which no
> mathematician seriously entertains.
No, it would only be omega-inconsistent, which is quite different.
Omega-inconsistency means that you can show that some proposition
P is false of each number but cannot establish that it is false
of all numbers.

>     I don't understand this.  Are you saying that "nonstandard number
> theory" proceeds from the assumption that PA is inconsistent?
GT shows that G is independent of the other axioms of PA, so we can
assume it (leading to standard number theory) or assume its negation
(leading to nonstandard number theory).  In the latter case, omega
inconsistency leads to the realization that 0,1,2,... are not all
the integers there are, and so there are nonstandard numbers
in such theories.

But if ~G had a proof in PA (and this cannot be ruled out, because
both G and ~G are consistent with PA), then there is no bifurcation:
~G is a theorem saying that G has a proof, but because of the
omega-inconsistency, we cannot construct such a proof!

>     Clearly, if PA is consistent, then there exist theories in which PA's
> G is false, PA+~G being an obvious example.  However, that theory would
> have its own G, which is undecidable in that theory.
Of course.

> > But if there were an *independent* proof of ~G, then nonstandard
> > number theory (analysis, topology, etc. etc.) is the only kind
> > there is, and nonstandard numbers must exist whether we want them
> > or not.
>
>     It what sense is analysis or topology "nonstandard number theory"?
No, I mean "nonstandard number theory, nonstandard analysis, nonstandard
topology, etc.".  Nonstandard number theory carries with it a
nonstandard theory of everything thing else mathematical, necessarily,
since all mathematicses are grounded on PA, as you say:

> Certainly, neither field is incompatible with Peano Arithmetic.  In fact,
> analysis is ultimately based upon PA, since the reals are based on the
> rationals which are based on the integers which are based on the naturals.
Just so.

>     For that matter, in what sense are you using "independent" and "exist"?
By "independent" I mean a proof which does not depend on Goedel's Proof
or meta-mathematical reasoning.

--
John Cowan                                   cowan@ccil.org
One art/there is/no less/no more/All things/to do/with sparks/galore
        --Douglas Hofstadter