Insanely Long Proofs

Azimuth

There are theorems whose shortest proof is insanely long. In 1936 Kurt Gödel published an abstract called “On the length of proofs”, which makes essentially this claim.

But what does ‘insanely long’ mean?

To get warmed up, let’s talk about some long proofs.

Long proofs

You’ve surely heard of the quadratic formula, which lets you solve

$latex a x^2 + b x + c = 0 $

with the help of a square root:

$latex displaystyle{ x = frac{-b pm sqrt{b^2 – 4 a c}}{2 a} }$

There’s a similar but more complicated ‘cubic formula’ that lets you solve cubic equations, like this:

$latex a x^3 + b x^2 + c x + d = 0 $

The cubic formula involves both square roots and cube roots. There’s also a ‘quartic formula’ for equations of degree 4, like this:

$latex a x^4 + b x^3 + c x^2 + d…

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