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Unformatted text preview: that 13 cannot be replaced by 2, but can it be
replaced by 3? This is open.
There is virtually no understanding of the nonnegative
(integral) parts of ranges of polynomials of several rational
variables with rational coefficients. It is well known that
they are r.e.
4. CONSTRUCTIVITY.
In mathematical logic, constructivity is treated in terms of
certain formal systems based on intuitionistic first order
predicate calculus which go back to Heyting. This is
definitely not a.a.g. friendly.
In many general contexts, the existence of a constructive
proof of a theorem implies a sharper form of that theorem. 9
That sharper form may be false or open. Great interest may be
attached to the sharper form, independently of any interest
in the general foundational concept of constructivity.
As a first example, consider the following well known fact:
*) For all polynomials P:Z Æ Z of nonzero degree, there are
finitely many zeros of P.
A constructive proof of *) would imply the also well known
sharper fact:
**) There is an algorithm such that for all polynomials P:Z Æ
Z of nonzero degree, the algorithm applied to P produces an
upper bound on the magnitudes of all zeros of P.
This can be seen to be a special case of the following
general principle.
Suppose that there exists a constructive proof of a statement
of the form
("n Œ Z)($m Œ Z)(R(n,m)).
Then there exists an algorithm a such that
("n Œ Z)(R(n,a(n))).
Now consider the obvious statement
#) For all multivariate polynomials P from Z into Z, $ a
value of P whose magnitude is as small as possible.
If #) has a constructive proof then the following sharper
statement must hold:
##) There is an algorithm such that for all multivariate
polynomials P from Z into Z, the algorithm applied to P
produces a value of P whose magnitude is as small as
possible.
But using Matiyasevich/Robinson/Davis/Putnam, one can refute
##). Hence #) has no constructive proof.
There are important examples in number theory where the
constructivity is not known. E.g., in Roth’s theorem about 10
rational approximations to irrational algebraic numbers, and
Falting’s solution to Mordell’s conjecture.
5. THE ACKERMAN HIERARCHY.
This is a basic hierarchy of functions from Z+ into Z+ with
extraordinary rates of growth. Yet these rates of growth
occur naturally in a numbe...
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 Fall '08
 JOSHUA
 Math, Logic

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