MathMeanMathLogic042100

An inter pretation of s1 into s2 consists of

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Unformatted text preview: rpretations between sets of free variable statements. An inter-pretation of S1 into S2 consists of definitions of the functions of S1 by free variable statements using the functions of S2 with the property that the translation of each statement in S1 through these definitions follows from S2 algebraically. THEOREM 8. Let S be a finite set of free variable state-ments which hold universally in some infinite algebra. There exists a free variable statement j such that S » {j} holds universally in some infinite algebra but is not interpretable into S*. Theorem 8 follows from the second incompleteness theorem. On the other hand, I don’t see how to derive the second incompleteness theorem from Theorem 8. 3. RECURSIVELY ENUMERABLE SETS OF INTEGERS. Recursively enumerable (r.e.) sets of integers occur throughout math logic. The most common definition is: There is an algorithm such that S is the set of all integers n for which the algorithm eventually finishes computation when applied to n. This very simple definition of course depends on having a model of computation. And there is a great deal of robustness 8 in that any reasonable model of general computation - without regard to resource bounds - will yield the same family of sets of integers. However, no one at the moment knows a really friendly way of defining what a “reasonable model of general computation” is. So for the purposes of a.a.g. friendliness, we avoid models of computation. We present a known characterization which comes from the solution to Hilbert’s 10th problem by Matiyasevich/Robinson/Davis/Putnam 1970. To begin with, r.e. sets of nonnegative integers are normally considered rather than of integers. S Õ Z is r.e. iff S « N and -S « N are r.e. The following is a byproduct of Matiyasevich/ Robinson/Davis/Putnam. THEOREM 9. S Õ N is r.e. iff S is the nonnegative part of the range of a polynomial of several integer variables with integer coefficients. From Matiyasevich 1992 con-cerning nine variable Diophantine representations, one can easily read off the following: THEOREM 10. S Õ N is r.e. iff S is the nonnegative part of the range of a polynomial of 13 integer variables with integer coefficients. 13 can be replaced by higher number. It is known...
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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