Re: help with Godel's

Nam D. Nguyen says...

Daryl McCullough wrote:
Nam D. Nguyen says...
I think it'd be "tough" to try it. One of the "dubious" feature of the
successor function, hence of any arithmetic model, is that once you either
prove or assume one exists, you could prove the existences of uncountably
many *others*. And it seems impossible to distinguish one from another.

Why do you keep saying that? It's not true. If you have any successor
function whatsoever, you can define a minimal model of PA in terms of
that successor function. That minimal model is *unique* (up to isomorphism).

Because you keep not understanding what I said: models involve
more than just sets, where you could apply the adjective "minimal"
to.

That's *all* that's at issue when the question is whether (for
example) GC is true. Any model whatsoever can be used to decide
that question: Take M to be any model model whatsoever. Take M'
to be the smallest submodel. Then if GC is true in M', then its
true of the standard naturals.

Model involve interpretation of n-ary relations corresponding
to the counterpart n-ary non-logical symbols. And for a given
non-logical symbol, there could be more than one relation, thus
more than one interpretation, thus more than one models, over the
very same set.

Sure, but the truth of Goldbach's Conjecture doesn't depend on
which of those interpretations you use.

For example, start with the real numbers R. Let our "successor function"
be defined by successor(x) = arctan(x). Let our "zero" be pi.
Then we can define in terms of our zero and successor, the subset
R'

R' = the smallest set containing pi such that
if x is in R' then arctan(x) is in R'

In terms of our zero and our successor, we can define
addition and multiplication. Then for this weird choice
of plus, times, successor, zero we can ask whether
Goldbach's conjecture is true. It will be true of
this structure if and only if it is true of the
standard naturals.

--
Daryl McCullough
Ithaca, NY

.

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