Re: help with Godel's

Daryl McCullough wrote:
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.

If one cares to look, there are always questions on the existence
of the naturals. For instance, which one of them is the natural
number 0? (There are uncountably many successor functions on them,
naturally!).


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.

Why then not *formally admitting* that, for the benefit of improving
reasoning framework, instead of keeping a silence for more that 70+
years, and still at it?


--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

  • Re: Model here, model there
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  • Re: help with Godels
    ... Daryl McCullough wrote: ... You can let absolutely any mathematical object be your zero, ... true of any structure isomorphic to the naturals. ... A standard model has no proper submodels. ...
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  • Re: Learning Logic and Set Theory
    ... Okay, according to another thread ("Metamathematically True or False?"), ... than sets of naturals. ... the concept of "expressing" a set generalizes to non-r.e. ... Daryl McCullough ...
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  • Re: Epistemology 201: The Science of Science
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  • Re: Epistemology 201: The Science of Science
    ... Daryl McCullough said: ... Allan hasn't pointed out any contradictions. ... > Cantor pointed it out in the case of rationals and naturals. ...
    (sci.cognitive)
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