Re: help with Godel's

herbzet says...

OK, fine. Do the naturals have an objective existence, or do
they exist only in the mind? Does there exist an actual infinite
number of naturals, or are they merely infinitely extensible?
Etc., etc.

I don't think that any of those questions have definitive answers,
but for most purposes, we can treat the natural numbers as if they
had an objective existence. I don't see how the difference matters
much.

Perhaps, as you say, the purpose of PA is not to define the
naturals, but to prove things about them. The question is
about the consistency of this proof-apparatus for natural
numbers, i.e., the existence of a model.

Well, to the extent that any model exists, a model of the
natural numbers exists. You can voice philosophical skepticism
about the whole idea of "model", but there is no *mathematical*
question about the existence of the naturals.

DM says "We point to the actual natural numbers". Where are
these actual natural numbers? In what sense are they actual?

That's a philosophical question that you can entertain yourself
with, but it has no mathematical relevance.

Is the idea here that the axioms of PA are consistent because
they are true of our conception of the naturals, a conception
which we _already know_ to be a consistent conception?

Yes.

If we already know it is a consistent conception, what is the
problem with an explicit accounting of that knowledge?

Nobody knows what that would even mean. Can you given an
explicit accounting of any knowledge whatsoever?

Shall we just deem that knowledge
as "intuitive", and let it go at that?

There is more to it than that. We have *experience* with the
natural numbers. We've had 2000 years or more of asking questions
about them and working hard to answer those questions. We've
done billions of calculations involving them. We've proved many
theorems about them. We've used them to do our science and engineering.
We have seen that the structure of natural numbers "hangs together"
in a very robust sense. There might be a sense in which none of
this is definite *knowledge* that the concept of "natural number"
is consistent, but to that extent, there is no definite knowledge
of anything at all.

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

  • Re: Model here, model there
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  • Re: Model here, model there
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  • Re: Model here, model there
    ... Conis true if and only T is consistent. ... to a number between 0 and 255, then just concatenate ... naturals, you come up with a formula Formthat is true ... Daryl McCullough ...
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  • Re: My talk about Godel to the post-grads.
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