Re: what makes it true?

Esa A E Peuha a écrit : 
Robert Low <mtx014@xxxxxxxxxxxxxx> writes:


No, they aren't. The second order axioms are categoric,
and all models of them are isomorphic. But the first order
axioms, which are the relevant ones here, are not categoric,
and admit non-standard models. If GC is not decidable, then
it is true in the standard model, but false in some (non-standard)
model.



Oh, right, it's the Dedekind-Peano structures that are isomorphic, not
all possible models.  Still, I don't quite see how a counterexample to
GC in any model wouldn't be a counterexample in every other model.

Easy : a counterexample is n (obviously "infinite", i.e > all standard integers, if GC is true) such that there is no primes p and q with p+q=n. At least one of p and q would be infinite, so at least their existence in the standard model for all n doesn't prove a thing (in fact, what "would happen is that, somehow, "infinite" proimes are so far apart in this region that n is never obtained as a sum. Of course, heuristic arguments show this to be quite unplausible, but if there was a way to show it is in fact impossible, it would give at once a proof of GC...
.

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