Posted on 02/24/2004 6:38:39 AM PST by dead
I've never claimed that human reasoning is a consistent, formal mathematical system with only finitistic methods of proof. That's the precondition for containing Godel statements.
Show me. ... don't bother with in-practice arguments like the limited human lifespan.
What, in principle, is an upper bound on the longest proof that could be written down and reasoned over? I can see several ways of getting at it. Because the universal expansion is accelerating, the amount of energy available for the calculation is bounded. There is some irreducible enery cost to preparing/erasing a bit of information. The amount of that needed will be at least proportional to the length of the proof. That gives an upper bound on the length of proof accessible to us.
Whatever that number is, there are theorems whose smallest proof is longer.
No, the nature of Gödel statements is that their truth or falsity is not provable by the set of axioms of the formal system in which they appear.
Yes, that's true. But the truth or falsity of any statement is only relative to some formal system. I was responding to the statement in your post which I took to imply that Godel statements could be said to be true or false irrespective of any formal system. IOW that mathematical statements can have some kind of objective truth.
No mathematical statement is undecideable in an absolute sense; one can always add an otherwise undecideable statement as an axiom (or its negation or some set that of statements that imply it or its negation) and presto, it is then decidable.
That's what's up with CH. Evidently it would be bad form to just add CH so they'll cook up something not quite so blatant instead.
In other words, there are proofs that require an infinite number of steps? That's the immediate implication.
I guess it'd help me if I understood whether we are talking about proofs in general or only about algorithmic "proofs." It's true that there are brute force proofs which take a huge number of steps to check every case, but of course that doesn't imply that more "elegant" solutions don't exist. The four-color theorem may be a case in point.
Close?
Your getting the flavor of it.....
Perfect. This article is pure nonsense. You have nailed it.
Correct me if I'm wrong here, but the Parallel Postulate isn't a Gödel statement, because geometry is exempt from Gödel's Incompleteness Theorem (because the axioms of Geometry don't incorporate the axioms for arithmetic on Natural numbers).
It is worthy of keeping in mind, it seems to me, that the essence of the Gödel's Incompleteness Theorem is that "there are some 'truths' about arithmetic of natural numbers we can't prove or disprove using only the axioms for arithmetic of Natural numbers..." The Parallel Postulate isn't a "truth" about arithmetic of Natural Numbers.
It also occurs to me that just because Gödel proved all formal systems that include the axioms of arithmetic of natural numbers will contain undecideable statements, it does not follow that ALL undecideable Mathematical statements are Gödelian in nature, the Euclidean Postulate being one counterexample.
As well you AND your flea-bitten friend, Plato the Platy, should!
We can 'understand' nothing about our 'world'. We can discover the cognitive and perceptual processes and limitations of ourselves. This is what our 'world' is. If it isn't prewired (into the brain and nervous system)we can't do it.
Just a minor quibble here: in a manifold, EVERY point has a "local Euclidean" region around it, not just some points. That is to say it is everywhere locally Euclidean, but globally non-Euclidean.
That's not proof; that's "decideability" by fiat. It reminds of a quote:
"Some men think the world is round, others think it flat. It is a matter capable of question. But if it is flat, will the King's command make it round, and if it is round, will the King's command flatten it?" - "A Man for All Seasons" -Robert Bolt.
That's what's up with CH. Evidently it would be bad form to just add CH so they'll cook up something not quite so blatant instead.
Let me guess, you don't like Mathematicians, do you?
;-)
I don't know. All I said was that the number of steps is unbounded.
I guess it'd help me if I understood whether we are talking about proofs in general or only about algorithmic "proofs."
All proofs in a formal system.
Holy cow, this is really amazing to me, VA. My inclination is to say that any proof of any theorem that never gets to the punch line is no proof at all. Surely such an important theorem has a name. Please tell me what it is so I can go look it up. Thanks. I mean, it's embarrassing; I have an advanced degree in mathematics and I've never heard of it! :-(
And a book of Vogon poetry.
-PJ
It seems to me that, if at the quantum level particles and waves are interchangeable (the duality as Hawking puts it), and then if particle physics is really string theory, then tell me what a wave would look like as a string counterpart, if a point on a wave is a particle counterpart.
-PJ
What I said was any given natural number. Unbounded and infinite are not the same thing.
I do understand that you're talking about any given natural number. Surely you understand that if there are theorems that require more steps (to prove) than any given natural number, then it follows that there are theorems that require an infinite number of steps. Of course, infinite and unbounded aren't the same in the case of convergence! But here you clearly have divergence.
Kindly just give me the name of the theorem you're quoting, please. I'd really like to take a look at it and can't find it on Google using the keywords and phrases I'd expect to find in any description or discussion of it. Thx.
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