I'm fascinated by Gödel's incompleteness theorems for a while now. The fact that a mathematical system can't be proven to be consistent or complete is amazing. But can that be false?
Could we construct/discover a mathematical system that is both complete and consistent and prove that within the same system? And in that system to prove that the current one that we are using (the Zermelo Fraenkel set theory or whatever) is inconsistent. And the inconsistent part of our current system to be the very proof of Gödel's theorems?
Can the following statement (or its proof) "No system can be proven consistent within itself" be the actual inconsistency in our system used today?
This is just a thought experiment that i need other opinions on.
Thanks!
Best Answer
Turning my comments into an answer:
First off, it's important to state the incompleteness theorem carefully (to be precise, I'm giving Rosser's improvement of Gödel's first incompleteness theorem):
Of particular note are the phrases "computably axiomatizable" and "interprets Robinson arithmetic." These are technical terms which I won't define here, but I will very briefly motivate them:
Computable axiomatizability is basically a "simplicity" condition, demanding that the theory in question not be so complicated that we can't even write down a set of axioms for it in a reasonable way.
The interpretability criterion is a "strength" condition. Our theory needs to be able to implement basic arithmetical calculations.
Basically, both too complicated theories (such as the set of all true sentences about $(\mathbb{N};+,\times)$) and too weak theories (such as Presburger arithmetic) are beyond the scope of applicability of the incompleteness theorem.
This isn't really related to your specific question, but I think being careful about what the theorem actually says will demystify it substantially.
OK, now on to the actual question.
There are two ways a theorem (including Gödel's) can be wrong: either the argument used is faulty, or the axioms used are not all correct. Here for simplicity I'm adopting a "naive Platonist" position, that there is a "true mathematics" out there even if we don't know what it is.
There isn't much to say about the first possibility; the incompleteness theorem's proof has been checked ad nauseam, including implemented on computers, and additional proofs have been discovered as well (see e.g. here). The interesting issue is the second possibility. Because of the rhetoric around the incompleteness theorem, it's quite reasonable to assume that some rather subtle axioms must be at work in its proof (note that I'm speaking of the axioms used in the proof of the incompleteness theorem itself, not the axioms of a system it applies to). This, however, is false: in fact extremely little is needed to run the proof! In order to doubt the incompleteness theorem, we have to be skeptical of rather basic principles of arithmetic itself - specifically, a very weak form of proof by induction. The logical structure of the proof of the incompleteness theorem is quite subtle, but the basic assumptions that argument begins with are just simple properties of arithmetic which we use all the time.