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It is quite well-known from Kurt Godel's (1931) ground-breaking result on theIncompleteness Theorem that rudimentary relations (i.e., those definable bybounded formulae) are primitive recursive, and that primitive recursivefunctions are representable in sufficiently strong arithmetical theories. It isalso known, though perhaps not as well-known as the former one, that someprimitive recursive relations are not rudimentary. We present a simple andelementary proof of this fact in the first part of the paper. In the secondpart, we review some possible notions of representability of functions studiedin the literature, and give a new proof of the equivalence of the weakrepresentability with the (strong) representability of functions insufficiently strong arithmetical theories. Our results shed some new light onthe notions of rudimentary, primitive recursive, and representable functionsand relations, and clarify, hopefully, some misunderstandings and confusingerrors in the literature.

On the Notions of Rudimentarity, Primitive Recursivity and Representability of Functions and Relations