The big question: how long is a piece of string theory? (can we ever understand the universe?)

Sydney Morning Herald ^ | February 25, 2004

[jpsb]

Is the author French?

[Virginia-American]

I don't know if it has a name; it's a consequence of there being an infinite number of theorems. I don't see how it follows that there are some that require an infinite proof.

Dr S: can you add anything?

[Doctor Stochastic]

It's been a long time since I looked at length of proofs. There's a long discussion of this (and lots of other things) here. I think that a proof using a Turing machine (and this covers all models of computation so far proposed) requires that the machine stop. There is an enumeration of all proofs though. Even if all proofs were finite, there would be infinitely many proofs in a system.

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.

[Virginia-American]

...requires that the machine stop. ...

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'...

[LibWhacker]

I don't see how it follows that there are some that require an infinite proof.

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.

[Virginia-American]

The number of theorems that can be proved is infinite, but each proof is finite.

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.

[LibWhacker]

Yes, quite right there, VA. I still think that one can show that the cardinality (if I can use that term) of some theorem must be "Aleph nought," the cardinality of the positive integers. Let me think about it some more.

[longshadow]

Something I've been wanting to point out before this thread withers away a dies:

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.

[Doctor Stochastic]

Good point.

[Doctor Stochastic]

Note that in your comments, you have proved that there are an infinite number of primes, not that there is an infinitely large prime.

(It's possible that) one may show that for any N, there is a proof that requires N+1 steps, but that doesn't mean that there is a proof that requires infinitely many steps. To show that, one must exhibit a statement that, for any N, cannot be proved in N or fewer steps.

[Doctor Stochastic]

I think that Hilbert's (and other people's) versions of proofs require that the proof be finite. This does not mean one cannot talk about infinite sets.

[LibWhacker]

Yep, I made two mistakes here. First, I thought it was trivial and intuitively obvious that if there were proofs of any given length N, then necessarily there were infinitely long proofs. Second, I thought that by showing there were infinitely long proofs that that would disprove the antecedent of the first, because the idea of an infinitely long proof is meaningless (wrong again!).

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.

[Doctor Stochastic]

This purported proof of the Catalan conjecture is generally rejected as valid because it requires an examination of infinitely many cases. In fact, the Catalan conjecture is still considered unproven (as of today.)

An example that some people consider to use infinitely many statements would be using induction. The infinitely long strings are telescoped by this idea. Example, to prove SUM(i,i=1,N)=N*(N+1)/2; first verify this is true for 1 (by direct computation); second show that the truth for N implies the truth for N+1, that is, SUM(i,i=1,N)+N=SUM(i,i=1,N+1) thus N*(N+1)/2+N = N*(N+1)/2 + 2*N/2 = (N**2+N +2N)/2 = (N+1)(N+2)/2. By direct calcualtion the formula for N implies the formula for N+1. The inductive procedure states that the result if true for all natural numbers.

The whole business is a bit hard. (At least Barbie thinks so.)