By Kripke, Saul
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Additional info for Elementary Recursion Theory and its Applications to Formal Systems
Often, the results of Gödel and Tarski are presented as if they involved all kinds of sophisticated ideas very different from these, but the inconsistency of the unrestricted comprehension scheme is essentially what proves those results. (Now, suppose that in the derivation of a paradox we used the unrestricted comprehension axiom with n parameters. How are we going to interpret 'x ∈ y' in this case? One way will be analogous to the reduction of satisfaction to truth that we saw for the case of RE.
However, as Cantor showed, we could not code all infinite sequences of numbers by means of numbers, so the satisfaction relation for formulae of RE cannot be represented as a relation between numbers. However, for our purposes it is really unnecessary to contemplate the full satisfaction relation. It will be enough to be able to define within RE the satisfaction relation restricted to finite assignments, or even a relation Sat(a,s), which holds between a (code of a) formula A and a (code of a) finite function s which assigns non-negative integers to all the (codes of) variables appearing free in A and satisfies A in the obvious sense (thus, if Sat(a,s), s need not be a sequence, for its domain need not be an initial segment of the non-negative integers -nor an initial segment of the codes of variables-, and s need not be an assignment, for it can assign values to things other than codes of variables).
But not recursive; and yet that set is clearly generated from a recursive basis set (the axioms) and recursive generating relations (the inference rules). Exercises 1. a) Prove that every k-place constant function is recursive. Prove that the successor function is recursive. ,mk) in k variables is recursive (partial recursive), so is any k-1 place function obtained from φ by identifying two variables. 2. a) Prove that the composition of two 1-place total (partial) recursive functions is total (partial) recursive.