Not quite.
On 7/9/07, Nollaig MacKenzie <[EMAIL PROTECTED]> wrote:
On 2007.07.09 11:47:42, you,
the extraordinary Jose Mario Quintana, emitted:
<...>
>
> To complicate matters, at least for me, there is a result by Tarski
implying
> that (an axiomatic version of) the field of real numbers is complete. I
am
> not familiar with its proof and I do not know how to reconcile it with
the
> former statement [that elementary number theory is incomplete].
<...>
Back in the day, some suggested that this might
seem less surprising if one considered Fermat's
"theorem":
There is no N > 2 such that x^N + y^N = z^N
You mean other than x,y,z == 0? Which is why the _exclusion_ is given
below?
x,y,z ranging over the integers >1. If there were a
complete proof procedure for elementary number
theory this would be easy to show true or false
("in principle" :-). For x,y,z ranging over real
numbers, it's obvious the "theorem" is false.
How about the generalization:
Summa(x[i]^N)[over i in I] = z^N [over x,z,n in R]
This becomes much more interesting, in that there ARE example son BOTH
sides.
And more interesting yet in the complex field.
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