3 ms·
Seems to me quite interesting that you need methods involving negative polynomial coefficients to prove statements that use only positive numbers. Actually, it
by brazzy 14y ago
Seems to me quite interesting that you need methods involving negative polynomial coefficients to prove statements that use only positive numbers.
Actually, it seems to be a concrete example for Gödel's incompleteness theorems: a very simple system can be used to make statements whose veracity can be proven only by using concepts outside the system.
- carstimon 14y agoSomething similar: If you want to write down a formula for solving a cubic: ax^3+bx^2+cx+d = 0, for give a,b,c,d, you can't avoid using complex numbers. This is true even if you only care about cubics which have real solutions. To clarify that last sentence, contrast it with the quadratic formula, which has no mention of the imaginary unit i. It's true that a quadratic with real coefficients might have only complex solutions, but you might just interpret that as having no solutions: once the formula tells you "take the square root of a negative", you know it has no real solutions. Maybe this could be worked into something similar to the high school algebra problem, but with a list of axioms about real numbers.
- visarga 14y agoThat's what I thought too: no fixed set of axioms can prove all the possible true affirmations. There is always going to be one that is outside of their scope. That makes mathematics more interesting, almost like a living organism, full of surprises.