[StevenXC]

Agreed..

> Therefore, all basic claims are more or less false. We just do not know why they are false. The entire body of science and mathematics therefore does not rest on a set of true statements but on a set of difficult-to-prove-false statements. Disproving a claim may even require showing where we can find a bug in the successor function. This is seriously hard, but still possible, because we know that these bugs must exist.

We do NOT know that these bugs exist. Gödel has shown us that any useful formalization of mathematics (such as the commonly utilized Zermelo–Fraenkel "ZF" set theory) cannot prove its own consistency. However, that does not imply that it must be inconsistant.

The fact that thousands of mathematicians have used ZF for about a hundred years without finding a contradiction makes me confident that even if it's inconsistant, we should be able to patch it up easily enough. But honestly I don't expect that to happen.

[gizmo686]

> The fact that thousands of mathematicians have used ZF for about a hundred years without finding a contradiction makes me confident that even if it's inconsistant, we should be able to patch it up easily enough. But honestly I don't expect that to happen.

As an example of this, the Greeks assumed that every number was rational (actually, that any two numbers could be expressed as an integer multiple of some common unit). When they did prove that this was false, they re-worked math without it and found that (almost?) all of their other theorems still worked even without the assumption that all numbers were rational.

[eriksank]

In my opinion, Kurt Gödel's claim points to something that seems to go wrong in Alonzo Church's successor function. For the sake of the argument, let's call that a bug. It would take a serious amount of work to point out how that would affect Zermelo–Fraenkel.

[gizmo686]

Care to elaborate on how Godel's theroum implies a bug in the successor function.