One can of course talk about infinte objects using only finitary means. For example, mathematical induction is only a schema for infinitely steps. Transcendental induction is also possible.
I'll look around on Google and in some texts and maybe have something to post later.
I still don't know how to pronounce adele. I found three spellings in different encyclopaedias: adele, adéle, and adèle; not helpful.
I would think so! Otherwise we're in the position of 'proving' Fermat's Last Theorem by an infinite exhaustive search.
Even if all proofs were finite, there would be infinitely many proofs in a system.
That's what I was getting at.
Re: pronouncing 'adele'
Thanks. I *thought* I'd seen it with both grave and acute accents. (Grave seems more French-like somehow).
Funny, I can *pronounce* 'restricted product of p-adic completions'...
The perveyors of Gödelian Uncertainty love to cast it like a wet blanket over all of Mathematics, and by extension onto physics; but here's the rub:
They never seem to grasp that just because there exists SOME truths about arithmetic of Natural numbers that can't be proven or disproven from within the system, it doesn't mean that the theorems we DO PROVE are in any way suspect!
So long as Mathematicians, and Physicists, restrict themselves to using only those Mathematical Theorems which HAVE been proven, they can never fall victim to Gödelian Uncertainty. An unproven theorem is still an unproven theorem, regardless of whether it's unproven because it's difficult (Fermat's Last Theorem), or because it is a Gödel statement. And, as long as we don't use unproven theorems, they have no effect on anything we are doing!
In conclusion, Gödel's Incompleteness theorem poses no practical impediment at all to most of science and Mathematics, which is why I consider those who raise it in this context to be either ill-informed or disingenuous.