Representation of processive functions in Robinson arithmetic ?
In connection with his so-called incompleteness theorem Gödel discovered the beta-function. The beta-function theorem is important for the representation of recursive functions in the concrete calcule ALPHA of Robinson arithmetic. The other features are composition and minimization of primitive recursive functions. Recursive functions are no part of Robinson arithmetic, but they are representable by certain formulae.
The concrete calcule LAMBDA of a natural number arithmetic with first-order logic has been defined by the author. It includes straight recursion and composition of functions, it contains a wide range of so-called compinitive functions, with processive functions far beyond primitive recursive functions. They include e.g. Ackermann's function and similar constructions. All recursive functions (that are obtained by minimization too) can be represented in LAMBDA . As long as there is no proof that all processive functions are minimitive recursive (recursive but not primitive recursive) one has the problem of representing them in concrete calcule ALPHA of Robinson arithmetic. As long as the challenge of such a proof is not met there is the conjecture that there are calculative functions that are not representable in Robinson arithmetic.
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