ABC Conjecture

Heard the news?  It's a big deal if the proof goes through.
Shinichi Mochizuki recently released a 500-page proof (split up in four sections) that claims to have solved the ABC conjecture:

 $\, \forall \, \epsilon > 1 \, \exists \,$ only finitely many coprime $a, b$ such that $d^{\epsilon} < c$ where $c=a+b$ and $d$ is the product of the unique prime factors of $abc$ (the "square free" part).

Apparently he needed to use non-well-founded sets to get this result. Neat stuff.
Paper 1     Paper 2     Paper 3     Paper 4
Updated to add better-educated folks' commentary on the paper:
Take one  Take Two  Take Three  MathOverflow