Re: help with Godel's

Nam D. Nguyen says...

Daryl McCullough wrote:

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.

To the extend we manage to *prove* something then we have a theorem
which is true in all models, isomorphic or not. But that has nothing
to do with certain arithmetical truth values that we don't yet know
(such as GC).

Yes, but the truth of GC certainly doesn't depend on what object
we choose to call 0 and what function we call "successor".

And to the extend that there are questionable arithmetical
truth value, there is legitimate skepticism about the particular
model we claim we could hold dear to us (namely the natural numbers).

I don't see why. There are questions that we don't know the answer
to, but that doesn't mean that there is any ambiguity about the
model we have in mind.

Given *any* model M of PA whatsoever, the standard model is isomorphic
to the smallest submodel M'. A standard model has no proper submodels.
So the question of "nonstandard models" has nothing to do with what we
choose as 0 and what we choose as successor. We can always make a standard
model for any such choice.

Oh, I simply suggest that we put what you said ("there is no definite
knowledge of anything at all [and of arithmetics in particular]") into
the framework of FOL, for instance.

I don't see why that is necessary. Logic is about what follows from
what. We may not know for certain what axioms are a satisfactory
foundation for mathematics, but we have a good idea of what *follows*
from whatever candidate axioms we choose.

--
Daryl McCullough
Ithaca, NY

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