Question

Is the following a theorem?

Every infinite recursive subset of $\mathbb{N}$ is the range of a strictly increasing primitive recursive function.

Answer 1

Let we have R.E (Recursive enumerable) set x. Since x is r.e. we have X = ran(f) for same recursive function f. I can find an infinite recursive subset s Subsetequalto X as follows. Each time see. f is enumerable some element of X longer than every element previously I put it in s The bigger than every element previously means that the set is in fact intuitively n elementof s. I wait until the process either puts n. into s (in which case n elementof s) or puts something bigger than n. into s (in which case n NotElementof s since we can never back to it). Now we have to apply our strategy define a auxiliary function I which counts when f goes up: i(0) = 0 i(n + 1) is the least x such that f(x) > f(i(n)) \r\n we could add the requirement that x > i(n) but by induction we can this is not necessary