Godel's Incompleteness Theorem & some philosophy

Mr. Flibble

I came across this in a lecture today.

Godel's Incompleteness Theorem basically states that you may be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing this you will create a larger system with its own unprovable statements.

So for instance, not all propositions in mathematics can be proven using the rules and axioms of mathematics.

This I'm OK with, but then I'm told that "A common misconception about the theorem is that it imposes some profound limitations on knowledge, science and mathematics".

I don't see this as a misconception, surely, if the theorem is correct, it does place limitations on knowledge. If we are in a system (life, the universe, everything & whatever) then we cannot prove everything within this system without leaving it and therefore creating another super-system, which itself cannot be proved, ad infinitum.


Am I missing something here?

I would have posted this in a Philosophy forum, but they would just talk in circles and wreck my head :rolleyes:

ecksor

I think you mean "every conceivable true statement" ? Similarly "true propositions" etc. Your description makes it sound like I can prove a false statement by this chicanery!

This I'm OK with, but then I'm told that "A common misconception about the theorem is that it imposes some profound limitations on knowledge, science and mathematics".

I don't see this as a misconception, surely, if the theorem is correct, it does place limitations on knowledge. If we are in a system (life, the universe, everything & whatever) then we cannot prove everything within this system without leaving it and therefore creating another super-system, which itself cannot be proved, ad infinitum.

I'm not familiar enough with this to say for sure, but I think the main thrust of this point is that although there is a theoretical limitation on what we can figure out to be true, it doesn't pop up as a practical limitation as often as people sometimes suggest it does. I.e, I don't think anyone has actually run into a brick wall in terms of mathematics research because of this. It merely shows why a project such as Russell's Principia isn't that useful.

Son Goku

Godel's Theorem basically says that with every set of mathematical axioms you get a few free statements whose truth you must decide on. It isn't what it first appears because what this really means is that even a perfect set of axioms don't give a unique system.

To explain:
For instance Set Theory is done today using the Zermelo-Frankel axioms.
Using these axioms you can construct a lot of mathematics, however you're eventually lead to something called the Continuum Hypothesis. Whether it is true or not can't be shown from the Zermelo-Frankel axioms. So you either accept that the Continuum Hypothesis is true or you don't, you can't prove it one way or the other.

So you get two different branches of mathematical systems, one that accepts the Continuum Hypothesis and one that doesn't.

Lets say you pick that it is true the fact that the Continuum Hypothesis is true becomes a new axiom.
However eventually you're led to another Hypothesis you have to accept or not and so on.

To sum up, every set of axioms will always generate questions whose answers have to be accepted as a new axiom.

Mr. Flibble

ecksor wrote:

I think you mean "every conceivable true statement" ? Similarly "true propositions" etc. Your description makes it sound like I can prove a false statement by this chicanery!

I think I do too. Or at least you can prove every statement to be true of false.


Ok, so what I understand by "profound limitations on knowledge, science and mathematics" must differ from others.

Do we agree that the theorem means that, in theory, we cannot answer everything about everything? I would call that a profound limitatiom, even if we are probably never going to hit that wall.

Son Goku

Mr. Flibble wrote:

I think I do too. Or at least you can prove every statement to be true of false.


Ok, so what I understand by "profound limitations on knowledge, science and mathematics" must differ from others.

Do we agree that the theorem means that, in theory, we cannot answer everything about everything? I would call that a profound limitatiom, even if we are probably never going to hit that wall.

It's more a statement about the nature of axioms as building blocks. I think it opens mathematics up more than it sets limitations.