Gödel's theorem in five words

Self-referential. Consistent. Complete. Pick two.

Comments

[Justin Sheehy]

Indeed.

But I think that one real brilliance in Gödel's approach on that topic was the realization that "expressive enough to use integers" forces you to have "self-referential", even if you really try hard to avoid it.

That's what seemed to upset Russell so deeply about this result.

[Justin Sheehy]

...which makes me wonder which ones Russell picked of the classic "fast, cheap, good" version of the saying. ;-)

[Joe]

Excellent. Thank you.

[mg]

Justin, I agree with you that the truly brilliant part of the theorem is just that; basic arithmetic alone is complex enough that its axioms can be fully represented within it.

My impression of at least Hilbert, if not Russell's, approach, was the belief that you could have all three: basic integer arithmetic, consistency, and completeness. This was the project of philosophical "formalism", and Gödel's work smashed that project to bits.

My restatement, on further reflection, is probably better characterized as MG's Corollary. Gödel's first theorem was: the self-referential, consistent system must be unable to express a true statement (namely, "This statement cannot be proven in this system.") and is therefore incomplete. Gödel's subsequent theorem was: the self-referential, complete system must prove the statement "This system is inconsistent." as well as the statement "This system is consistent." I believe my whimsical characterization follows naturally from these.

(And yes I am aware that I am grossly simplifying but I think after a great deal of study it is defensible to draw these abstract descriptions from the concrete logical proofs.)

I am really interested in propositions like this: "A, B, or C -- pick two." As Justin alludes, the famous "Good, fast, cheap" is another one of these. Are there others? I wonder. I imagine I could draw this as a set-theoretic diagram, but I have no idea how to express this idea purely mathematically/logically. It seems like a powerful and compelling construction. Very intriguing.