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Unformatted text preview: rpretations between sets of free
variable statements. An interpretation 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 statements
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 concerning 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.
 Fall '08
 JOSHUA
 Math, Logic

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