Posted on 02/24/2004 6:38:39 AM PST by dead
Dr S: can you add anything?
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 reasoning is similar to the proof that the number of primes is infinite. You start by assuming that there is some largest prime and show that that leads to a contradiction. We can do the same thing here . . . Assume that there is some "largest proof," and you'll see this leads to an immediate contradiction.
If assuming that there is a longest proof leads to a contradiction, then there is no longest one. That doesn't imply that any of them is infinite; by analogy, no prime number is infinite.
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.
But in trying to prove it, I found out in doing a little research that not only are there proofs that are infinitely long, their existence contradicts nothing! Some mathematicians seem quit comfortable with the notion. This is all covered in a field called "proof theory," to which I had no exposure in grad school, and I gather is an area of mathematical logic.
It still seems very strange to me, though, and by "proof" they must be working with some generalized definition of "proof." Heck, I can even think of a specific example of a countably infinite proof now, though I have no idea if this is what they are talking about, but I think it must be. For example, take Catalan's conjecture that says 8 and 9 (23 and 32) are the only consecutive powers in Z+:
Proof
1 and 2 aren't consecutive powers
2 and 3 aren't consecutive powers
3 and 4 aren't consecutive powers
4 and 5 aren't consecutive powers
5 and 6 aren't consecutive powers
6 and 7 aren't consecutive powers
7 and 8 aren't consecutive powers
8 and 9 ARE consecutive powers
9 and 10 aren't consecutive powers
10 and 11 aren't consecutive powers
11 and 12 aren't consecutive powers
. . .
And so on, forever. Q.E.D.
This "proof" has a countably infinite number of lines and if such infinitely long proofs are allowed, is in fact a "proof" since it exhausts all consecutive pairs -- PROVIDING, of course, that the conjecture is true (I suppose I should've used an example whose truth is known, but I'm too lazy now to go back and change it).
I even found references to artificially expanding any finite proof so that it has an infinite number of lines, and now that I've read a little bit about proof theory, I wouldn't doubt at all that it's possible and wager that if a person tried, it wouldn't be too hard to come up with an example of that also.
Finally, I found a claim that there exist proofs that are not only infinitely long, but have infinite logical "depth;" i.e., that are hopelessly beyond man's understanding. This is bad news, imo. Never dreamed there were results like this out there. Nice talking to you guys.
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