Diagonalization in double frames

Logica Universalis 4 (1):31-39 (2010)
Abstract
We consider structures of the form (Φ, Ψ, R ), where Φ and Ψ are non-empty sets and is a relation whose domain is Ψ. In particular, by using a special kind of a diagonal argument, we prove that if Φ is a denumerable recursive set, Ψ is a denumerable r.e. set, and R is an r.e. relation, then there exists an infinite family of infinite recursive subsets of Φ which are not R -images of elements of Ψ. The proof is a very elementary one, without any reference even to e.g. the -theorem. Some consequences of the main result are also discussed.
Keywords Diagonalization  double frames  incompleteness
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