I don’t have the chops to follow the details of Gödel’s famous Incompleteness Theorem, but thanks to Ernest Nagel’s impressively accessible Gödel’s Proof I think I get the basic idea, which goes like this:
- Gödel comes up with a way of representing mathematical proofs as sequences of integers. A thorny bramble of number theory ensues. Each inference gets mapped to a prime. Or something. I forget. This is where the heavy lifting happens.
- Having established this scheme, Gödel uses it to encode a tricky self-referential statement that ultimately boils down to something like “This theorem is false”. If it’s true, then it’s false, and if it’s false, then it’s true.
- The most reasonable thing to do in this situation is to throw up your hands and say that the truth of this particular statement cannot be determined. Therefore within any axiomatic system there exists at least one theorem whose truth is forever undecidable.
I buy this. I even kinda understood it a couple years back when the Nagel book was still fresh in my mind, but then as now the question nags me, so what? As far as I can tell, Gödel’s theorem just refers to this particular statement. It doesn’t demonstrate that we’ll never be able to prove Goldbach’s Conjecture, or Fermat’s Last Theorem, or pick your favorite currently unresolved mathematical question. Furthermore, it seems a simple matter to amend the definition of “axiomatic system” to “a system that excludes a restricted class of willfully perverse party-trick paradoxes”. How is the Incompleteness Theorem not like the joke about the guy who goes to the doctor and says, “Doc, it’s hurts when I do that,” so the doctor says, “Don’t do that”?
This isn’t a rhetorical question. I assume people wouldn’t make such a big deal about Gödel’s theorem if it wasn’t actually a big deal. I just plain don’t understand why.