How did your mind came to apprehend this from your understanding of
the naturals? Do you believe in Church's thesis?

Indeed not everything is provable in PA. So how come we are absolutely
certain that PA is consistent? How can there be certainty without a
proof? If we can be certain about a mathematical truth without a
proof, what are the proofs for?

No, but I am asking you how you know that it cannot?

I am asking if those manifest truths are analytic, synthetic a priori,
or synthetic a posteriori?

And you still need a proof of consistency to make sure your
formalization of the meaning of the terms is correct.

So if you you somehow arrived at the conclusion that the axioms are
consistent, and at the same time you proved that consistency is
unprovable you need to explain how exactly you arrived at the
conclusion that they are consistent.

Could you please try to understand what Godel's theorem says,
before you start making grandiose claims about what it means?

On 2008-04-24, in sci.logic, Jesse F. Hughes wrote:

As far as I recall, Russell didn't claim that the axioms of
mathematics are analytic, but rather that statements of the form
"The axioms entail P" are analytic.  Thus, the question I asked
about choice just doesn't come up.

Russell certainly thought that the basic axioms of mathematics should
be reduced to purely logical principles. It was because he saw no way
of justifying the axiom of infinity and the multiplicative axiom as
logical truths he considered the project of Principia Mathematic as
having failed.

On 2008-04-23, in sci.logic, Newberry wrote:

Indeed not everything is provable in PA. So how come we are absolutely
certain that PA is consistent? How can there be certainty without a
proof? If we can be certain about a mathematical truth without a
proof, what are the proofs for?

We have any number of perfectly good proofs of the consistency of
PA. All of these necessarily involve mathematical principles not
contained in PA.

Your questions are, as Daryl notes, interesting, in the general
philosophical sort of way, but not in any way connected to any notion
about the paradoxical nature of Godel's results.

--
Aatu Koskensilta (aatu.koskensi...@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Newberry says...

On Apr 23, 7:24=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:
Newberry says...

Probably the simplest way to put is this: We are certain that PA is
consistent. We also know that we cannot prove it.

It's not that *we* can't prove it. Godel's theorem shows that
*PA* can't prove it.

No we cannot prove.

If you say so, but that's not what Godel's theorem says.

So your comments are not relevant to the issue of whether
Godel's theorem is paradoxical.

--
Daryl McCullough
Ithaca, NY

On Apr 23, 7:36 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
All the theories that Newberry says are mathematics are analytically
true (aside from geometry).  We know they are analytically true
because Newberry says so.

Obviously, I have no reservations about that thesis.

Also Hume, Frege and Russell said it but that is not the reason that
mathematics is analytic.

In any case the idea that mathematics is Platonic is pretty
incredible. Even if Goedel's result is not paradoxical it leads to
this extraordinary conclusion, which is a good enough reason to
reform PA.

As far as I recall, Russell didn't claim that the axioms of
mathematics are analytic, but rather that statements of the form
"The axioms entail P" are analytic. Thus, the question I asked
about choice just doesn't come up.

On 2008-04-24, in sci.logic, Jesse F. Hughes wrote:
As far as I recall, Russell didn't claim that the axioms of
mathematics are analytic, but rather that statements of the form
"The axioms entail P" are analytic. Thus, the question I asked
about choice just doesn't come up.

Russell certainly thought that the basic axioms of mathematics should
be reduced to purely logical principles. It was because he saw no way
of justifying the axiom of infinity and the multiplicative axiom as
logical truths he considered the project of Principia Mathematic as
having failed.

There is a contradiction right here. "perfectly good proof" and
"involve mathematical principles not contained in PA."

On Apr 23, 7:24=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:
Newberry says...

Probably the simplest way to put is this: We are certain that PA is
consistent. We also know that we cannot prove it.

It's not that *we* can't prove it. Godel's theorem shows that
*PA* can't prove it.

No we cannot prove.

On Apr 24, 12:36=A0am, Aatu Koskensilta <aatu.koskensi...@xortec.fi
wrote:

We have any number of perfectly good proofs of the consistency of
PA. All of these necessarily involve mathematical principles not
contained in PA.

There is a contradiction right here. "perfectly good proof" and
"involve mathematical principles not contained in PA."

But we utilize Goedel's theorem in the argument showing that we
cannot prove it.

On Apr 24, 7:36=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:
Newberry says...

On Apr 23, 7:24=3DA0am, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:
Newberry says...

Probably the simplest way to put is this: We are certain that PA is
consistent. We also know that we cannot prove it.

It's not that *we* can't prove it. Godel's theorem shows that
*PA* can't prove it.

No we cannot prove.

If you say so, but that's not what Godel's theorem says.

I did not say that Goedel's theorem says that we cannot prove it. I
said we cannot prove it. But we utilize Goedel's theorem in the
argument showing that we cannot prove it.

Therefore Goedel's result is paradoxical.