matt grime said:
a logician might go a bit potty of the consistent/complete Goedel's theorem discussion
I wouldn't consider myself a logician, but yeah
Godel discovered a property of any logical system that truly astounded mathematicians. He began by thinking about the way rules can be used to make statements. What Godel found was that if these rules contain no contradictions then there is something strange about the statements that can be made with them: certain statements cannot be proved true using the available rules, even though they are true. Instead extra rules are needed to prove the point. So the original set of rules must be incomplete.
Godel’s theorem is that if a set of rules are consistent, they are incomplete.
isn't correct. Someone should send the artist a picture of Gödel's
Completeness Theorem. Hurkyl's given several nice explanations of Gödel's First Incompleteness Theorem here if anyone's curious.
Actually, now that I think about it, I wonder what exactly makes statements like those beautiful. For example, there is something beautiful to me about
1) \Phi \models \phi \Leftrightarrow \Phi \vdash \phi
It's certainly not because I think the font is pretty (though I suppose that might have some effect). At first, I think it's a combination of the statement's meaning and the simplicity with which it is stated. But I could state the same thing even simpler by saying
\clubsuit =
df (1).
But this doesn't make
2) \clubsuit
beautiful -- or at least not nearly as much so as (1). Stating (1) in a more complex way also eventually removes at least some of the beauty -- all of the statements preceding (1) in the chapter together basically say what (1) says. Even its closest translation into English isn't as satifying.
3) An L-formula is a logical consequence of a set of L-formulas if and only if that L-formula is deducible from that set of L-formulas.
So it seems the relationship between form and meaning that gives rise to beauty isn't so simple, even in math and logic. Anywho, I just think that's interesting; the same thing is at work in poetry (in natural languages).
