FOM: Simpson writes

[JSHIPMAN@bloomberg.net]

>Could you please remind me what Faltings' theorem says?
If the soltution set of a polynomial p(x,y)=0 has more than one
hole, then there are only finitely many rational solutions.
Reply:
This isn't satisfactory as stated, it is in no way clear to the
uninitiated what it means for a solution set to have holes.
I'd be willing to concede the point that Faltings' theorem is
as foundational as Matiyasevich's if "solution sets have >1
hole" could be replaced by a condition on p(x,y) whose meaning
was intuitively clear (even if it was a less general condition).
-- Joe Shipman