These two forces are put into conflict by the fact that given a model with a recursive coding on the natural numbers, any formula with bounded quantifiers and finitely many parameters will produce a recursive set of natural numbers consisting of the codes of the elements for which the formula holds.
12.
A formal parsing system G = ( V, & mu;, T, R ) consists of two finite disjoint vocabularies, V and T, a many-many map, & mu;, from V onto T, and a recursive set R of strings in T called syntactic sentence classes . ..
13.
If the elements of M were to be coded onto the ( standard ) natural numbers in such a way that the addition and multiplication operations of the model were recursive functions on the codes, then every subset of the model defined by a first-order formula with finitely many parameters and bounded quantifiers is a recursive set of natural numbers.
14.
Many degrees with special properties were constructed : " hyperimmune-free degrees " where every function computable relative to that degree is majorized by a ( unrelativized ) computable function; " high degrees " relative to which one can compute a function " f " which dominates every computable function " g " in the sense that there is a constant " c " depending on " g " such that " g ( x ) < f ( x ) " for all " x > c "; " random degrees " containing limit-recursive sets.
