Originally posted by ChronicLeaky
Goedel's theorems require the axiom system to have a specific property for that result to hold. They actually assert that if an axiom system is sufficient to construct the natural numbers, then either there is a statement derivable from the axioms whose negation can also be derived, or there is a statement for which neither itself nor its negation can b ...[text shortened]... whether the "axiom systems" in question need contain a construction of the natural numbers.
I thought that Gödel's theorems was proved? That the theorem was therefore correct? No doubt about it? (Even if the matematicians of that time didn't like it and that Gödel committed homocide of the entire mathematical world. Nothing in maths would therafter considered as true? However, Mathematicians have recovered after that blow.)
Let's take the example of natural numbers and it's axioms. The whole foundations lies on a set of axioms. Everything above this is based of these axioms. One of them is "There is a smallest natural number." It cannot be proven, we hold that for true without being able to prove it. Everything else has to rely that this axiom in particular is true. If it is not true, we have to start from the beginning again.
(Some says that 1+1=2 is self evidentory, but it's not. It relies on the axoms as well.)
However, if we come to try to prove that there isn't a smallest natural number, and we succeed, then this proof in turn have to rely on a more basic axiom, which by definition is not proovable.
For short: All mathematics begin with a series of axioms that cannot be proven, but self evident in its nature. If this self evidentiary is concidered be based on faith, then I agree with everyone, even if this belief is of a non religious kind.
Now, am I right in this, or am I only ranting?
Edit: Yes, there are other ways to define natural numbers, set theory provides with an alternate one.