A subset of Zn whose non-computability leads to the existence of a Diophantine equation whose solvability is logically undecidable
Abstract
For K ⊂eq C, let Bn(K)=(x1,...,xn) ∈ Kn: for each y1,...,yn ∈ K the conjunction (∀ i ∈ 1,...,n (xi=1 => yi=1)) AND (∀ i,j,k ∈ 1,...,n (xi+xj=xk => yi+yj=yk)) AND (∀ i,j,k ∈ 1,...,n (xi*xj=xk => yi*yj=yk)) implies that x1=y1. We claim that there is an algorithm that for every computable function f:N->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any integer n>=m(f), and returns a tuple (x1,...,xn) ∈ Bn(Z) with x1=f(n). We compute an integer tuple (x1,...,x20) for which the statement (x1,...,x20) ∈ B20(Z) is equivalent to an open Diophantine problem. We prove that if the set Bn(Z) (Bn(N), Bn(N 0)) is not computable for some n, then there exists a Diophantine equation whose solvability in integers (non-negative integers, positive integers) is logically undecidable.
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