Originally Posted by
Singu
That's because you didn't understand the Godel's Incompleteness Theorem properly, or you misunderstood how it works.
This is what the Theorem says:
1. In any sufficiently complex system, a system is either incomplete or inconsistent.
2. Such a system can't prove its own consistency ("Consistent" in this sense means that you can not prove both a statement and its opposite to be true, as it would be contradictory e.g. you can't prove that both unicorns exist and unicorns don't exist).
You're talking about #2. But #2 is just an extension or the consequence of the #1, and it's not the main point.
--
What Godel did was through a very complicated process, he has devised a way to prove any statements to be either true or false. Let's call it the function Provable(x), that can prove any statement to be either true or false. He has also devised a way to convert any statements into numbers, just as computer codes can be converted into 1's and 0's these days (this is called the Godel-numbering), so that we can simply plug in the numbers to the Provable(x) function. But instead of using numbers, we'll just use English letters for now because it's easier for us to understand. But in principle, it's possible to use numbers and thereby making it a mathematical statement.
He has also created the opposite of Provable(x), the NOT Provable(x). Let's call it the function NotProvable(x). Since this is just the opposite of Provable(x), it just reverses the answer. If NotProvable(x) = true, then Provable(x) = false, and vice versa.
This is all very good and all, but what can we do? Godel has done all this just so that he can prove a very important point.
Let's just suppose that NotProvable(x) is false. We want to say that NotProvable(x) was proven to be false by plugging it in to the Provable(x) function. What will happen then?
If NotProvable(x) = false
(To get a better picture, NotProvable(x) is like saying in plain English, "This statement is false". If we say that it's false, then it's true.)
So this results in NotProvable(x) being true, even though we said it was false? This is a logical contradiction and an inconsistency, since we have just proved that something that was supposed to be false, is true. That means we can prove a false statement to be true. This will be the end of math as we know it.
We don't want that to happen, so let's assume that NotProvable(x) is true instead. What will happen?
If NotProvable(x) = true
This is a better option, but it also means that NotProvable(x) can't be proved to be either true or false, since we can no longer use the Provable(x) function. It will forever remain a mystery.
We are only supposing that NotProvable(x) is true, only because we don't want an inconsistent system. There is no "proof" of this anywhere, in fact we have brought this "NotProvable(x) = false" from outside of the sytem. It is true, but unprovable. Therefore, the system is incomplete. There is a statement that can't be proven within the sytem.
And that is why a system is either incomplete or inconsistent. It also can't prove its own consistency, because if we did prove its own consistency, then as in the "NotProvable(x) = false" example, it will prove to be inconsistent. But we'd want a math system to be consistent, so we are only supposing that it's consistent, but also remaining incomplete forever (we can no longer prove it). Since it's not provable, we'd have to keep adding new axioms from outside of the system forever.