Gödel's incompleteness theorem?

Question

Goldstein described it as "In any formal system adequate for number theory there exists an undecidable formula -- that is, a formula that is not provable and whose negation is not provable"

Professor Ichi describes it as: "A retarded waste of time"

Why bother posing absolutely improvable questions: if you can't get to an answer yourself why do people feel like sharing the suffering...

The world of Maths is full of geniuses; but we are almost all completely nuts: its things like this that do our heads in, can everyone stop now with the conjectures and the theorems: what do you think...

Answer 1

I think that the secret to life is a good cup of coffee and a cigarette.

Besides, if it was easy **any** moron could do it. As we used to say at CalTech, "Where there is no confusion, there can be no prestige."

Therefore: "If you can't dazzle 'em with analysis, baffle 'em with topology." ☺


Doug

Answer 2

In the late 19th and early 20th century, there was a lot of interest in the foundations of mathematics. In particular, there was a desire to show that the axioms for arithmetic were both "consistent" and "complete."

Godel's Incompleteness Theorem basically ended this line of exploration, showing that there was no way for an axiomatic system complex enough to include number theory could be both consistent and complete.

Godel's theorem doesn't "pose improvable questions," it just shows that such questions exist. The point isn't to frustrate people, it's to answer a specific question about mathematical logic.

Can you trisect an angle with a straight edge and compass alone? That was a long-unsolved problem which turned out to be answered in the negative. It was only with the development of Galois theory that it was impossible to do it. Countless generations of mathematicians tried to trisect angles, square the circle, square the cube, etc. with straight edge and compass alone, and all failed because Galois theory showed it was impossible. Does that make the problem just irritating? I don't think so.