On Rudimentarity, Primitive Recursivity and Representability

Saeed Salehi


It is quite well-known from Kurt G¨odel’s (1931) ground-breaking Incompleteness Theorem that rudimentary relations (i.e., those definable by bounded formulae) are primitive recursive, and that primitive recursive functions are representable in sufficiently strong arithmetical theories. It is also known, though perhaps not as well-known as the former one, that some primitive recursive relations are not rudimentary. We present a simple and elementary proof of this fact in the first part of the paper. In the second part, we review some possible notions of representability of functions studied in the literature, and give a new proof of the equivalence of the weak representability with the (strong)  representability of functions in sufficiently strong arithmetical theories.

AMS Subject Classification: primary 03F40; secondary 03D20, 03F30

Słowa kluczowe: the incompleteness theorem, bounded formulas, rudimentary relations, primitive recursive functions, primitive recursive relations, representability

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