By Kripke, Saul
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Extra resources for Elementary Recursion Theory and its Applications to Formal Systems
Sets of formulae. , Rk among formulae. e. e. e. e. The Generated Sets Theorem is known to all logicians, although it is rarely stated explicitly. e. (and hence that some total functions are recursive) than primitive recursion. Of course, it does not provide a general method of proving recursiveness, but it is infrequent in mathematical arguments to have the need to show that a set or relation is recursive besides being recursively enumerable. It is usually emphasized as a basic requirement of logic that the set of formulae of a given language must be decidable, but it is not clear what the theoretical importance of such a requirement is.
Elementary Recursion Theory. Preliminary Version Copyright © 1996 by Saul Kripke ∨ (m = 0 ∧ n = 0∧ Pr(p)) is equivalent to Digp* (this remark is due to John Barker). To see this, suppose first that Digp*(m,n,p), m
0, then m must be an intermediate or final digit of n, so suppose z1 = 0.
E. ) This proof uses an idea due to Cantor. As we will see, once we have this theorem, Gödel’s first incompleteness theorem is just around the corner. The fact (if it is one) about the intuitive notion of computability corresponding to the enumeration theorem is that there is a semi-computable relation that enumerates all the semicomputable sets. It follows from this that there is a semi-computable set that is not computable. However, to prove the enumeration theorem for semi-computability, and thus to prove that not all semi-computable sets are computable, it seems necessary to use Church's Thesis.