Arithmetical Sets and Retracing Functions

Abstract

in Manchester (England) Retraceable sets and retracing functions were introduced by DEKKER and MYHILL [3] and have been further investigated in [l], [2], [lo], [ll], [12], and [18].Two questions were left open in [3], the second of which we answered in [HI. The first question was concerned with characterising retracing functions amongst the partial recursive functions, and one of our purposes in the present note is to show that in a sense no interesting characterisation can exist. For) it is easy to prove (Theorem 1 below) that the property of being a retracing function is complete for 2: predicates2); it follows from theorems of KLEENE [S], [9] that, for example, the retracing functions cannot be characterised as the partial recursive extensions of elements of any class of partial recursive functions all of which retrace an infinite hyperarithmetical set. I n particular, (I) there are retracing functions that retrace no infinite hyperarithmetical set, although by GANDY'S basis result [5] every retracing function retraces an infinite set of hyperdegree less thanIn contrast with (I), a simple generalisation of the KREISEL-SHOENFIELD basis result [15] implies (11) that every finite-one retracing function retraces an infinite arithmetical set, in fact an infinite set of degree less than O(') and so expressible in both 3-(number) quantifier forms. These results (I) and (11) give a different proof of our theorem [lS] that there is a retracing function that is not an extension of any finite-one-retracing function. Our other results are mainly refinements of various theorems in [3] and [lS]. I n [18] we constructed a finite-one retracing function that retraced no infinite recursive set; we show below that in fact a finite-one retracing function need retrace no set expressible in a 2-(number) quantifier form, so that (11) above cannot be improved in this direction.We recall that a set R is retraceable if there is a partial recursive function e such that e(ro) = ro and e(T,+l) = r, for all n , where ro, r l , . . . , are the elements of R in ascending order. Any such function is said to retrace R, and a partial recursive function is called a retracing function if and only if it retraces a t least one infinite set. If e ( x ) 5 x for all x in the domain of e then e is said to be downward; if in addition its domain contains its range and its range is infinite then it is said to be special. Finally, if a special partial recursive function is finite-one and maps only finitely many numbers onto themselves then it is called basic; every basic function is easily shown, using We denote the lowest hyperdegree by 0, and the hyperdegree of complete 2: and II: predicates by 0:".13 Ztschr. f. math. Logik

Π01-classes and Rado's selection principle

“…We should like to remark that a result similar to Corollary 5.1 was obtained by Yates [16,Theorem 7]. An analogue of Corollary 5.2 was obtained by Yates as Theorem 6 of the same paper.…”

Ramsey's theorem and recursion theory

“…If P is chosen so that H(P) has no member recursive in O' (Theorem 3.1), then T easily yields a finite-one retracing function which retraces no infinite set recursive in O'. Such a retracing function was shown to exist by Yates [13,Theorem 6]; another proof is given in [5,Theorem 4.22]. §5.…”

Supersimple sets and the problem of extending a retracing function

“…We The construction of I from k is very easy: Comment. It is known ( [5,Theorem 5.2]; [4,Lemma 3]) that if J is the unique infinite set retraced by some partial recursive, finiteto-one function, then J has degree of unsolvability ^ 0'. So, in particular, the set R of Theorem 5 has degree ^ 0'.…”

“…Hence J = {λ n \ n e N} is the only infinite set retraced by g. It remains only to prove that there exists a finite-to-one partial recursive restriction g' of g which retains the property of retracing /. The existence of such a function g', however, follows from [5,Theorem 8], since it is easily verified that {X n \neN} has degree of unsolvability ^ 0' (Simply note that λ« is a recursive function of n and s if we set X s n = 0 whenever Λ n is unattached at the end of Stage s.) That completes the proof of Theorem 1*.…”

“…Certainly, ζ retraces H among (possibly) others. Hence by [5,Theorem 2], I has 492 T. G. MCLAUGHLIN an infinite (retraceable) subset of degree strictly less than 0". But therefore also / has degree <; 0", since I is retraceable and so is reducible to all of its infinite subsets.…”