Re: help with Godel's

Nam D. Nguyen says...

Daryl McCullough wrote:

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!).

The structure of the naturals is only defined up to an isomorphism.
You can let absolutely any mathematical object be your zero, and you
can let absolutely any function that is 1-1 but not onto be your
successor function. What we prove about the natural numbers is
true of any structure isomorphic to the naturals.

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?

I'm not sure I understand what you are suggesting, but I haven't
kept silent about anything for 70 years (not yet, anyway).

--
Daryl McCullough
Ithaca, NY

.

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