MathMeanMathLogic042100

There is virtually no understanding of the

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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, con-structivity is treated in terms of certain formal sys-tems 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 produ-ces 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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