FOM: Faltings' theorem isn't basic

[Stephen G Simpson]

Lou van den Dries writes:
 > Well, what is "basic" for Steve seems to change from one email to the
 > next. In an earlier email he says:
 > 
 > "Matijasevic's theorem is about objects of a much more basic kind:
 > polynomial equations with integer coefficients"
 > 
 > and later:
 > 
 > "Matijasevic's theorem may be called foundational while the Lang
 > conjectures may not".

Where have I changed my mind about what's basic?

In context, the gist of the first statement was: (1) the subject of
Matijasevic's theorem is polynomial equations with integer
coefficients, (2) the subject of Lang's conjectures is rational points
on algebraic surfaces, (3) polynomial equations with integer
coefficients are much more basic than rational points on algebraic
surfaces.  

The second statement says: (4) Matijasevic's theorem is foundational
while the Lang's conjectures aren't.  (1)-(3) are part of the reason
for (4).  There's no inconsistency here.

 > Now, if the difference between the basicness of Matijasevic and
 > Faltings (I mean, between the questions addressed by their theorems)
 > is that Matijasevic considers integer solutions and Faltings considers
 > rational solutions, 
 ...

That's not the principal difference that I had in mind.

(1)-(3) are not the whole reason for (4).  The rest of the reason is
that Matijasevic's theorem asserts algorithmic unsolvability of the
most immediately obvious problem concerning the objects in question.
It's a striking example of algorithmic unsolvability of a rather basic
problem.  In this respect it's almost in a class by itself.  The Lang
conjectures, and even Faltings' theorem, are not nearly so neat, and
not nearly so striking to nonspecialists.

Of course mathematics contains some objects that are still more basic,
e.g. the natural numbers.  And it would be nice to have results that
would be even better than Matijasevic's theorem, by asserting
unsolvability of a striking problem dealing solely with these more
basic objects.  This is a foundationally important open question.

-- Steve