PCA Well-Orderings of the Line

There is a PCA well-ordering of the real line if and only if there is a real from which every real is constructible. The proof given below of the above statement is formalizable in ZFC; indeed, in second-order arithmetic. Despite this, it is open whether this result can be refined by considering the codes for PCA well-orderings. Specifically, let H be "there is a l well-ordering of wa if and only if w'0 c L." It would be particularly satisfying to have a proof in ZFC of H. More generally, for f E we let H(f) be "there is a well-ordering of wA that is 192 in f if and only if b c L(f)." We conjecture that (Vf)(H(f)) is provable in ZFC. Throughout the paper we will work in the space wA rather than the continuum, and use the logical notation Z instead of PCA. Throughout, g, 1h will be used as variables over wam. Let L(f) be the class of sets constructible from f. The 21degrees are the equivalence classes of wAd under the equivalence relation (f is \2 in g and g is Al in f). The first ordinal not Al in f is denoted by >2(f). For K c let AL(K) be UKL'(f). Let (f, g)(2n) = f(n), (f, g)(2n + 1) = g(n). Let (f, g, h) = ((f,g), h). Kis Turing closed iff (f ge K,h < T(f, g))heK. In Friedman [2], a complete discussion of the existence of minimal and minimum upper bounds for sequences of A4-degrees is given. Although perfect set forcing is never mentioned there, the techniques may be viewed as forcing arguments; i.e., where the conditions, for example, are the Al perfect trees, and in which 2 statements are forced if and only if they are true of all nonconstructible paths. The key theorem (Theorem I) of Friedman [2] is the key lemma of this paper. The following is the relevant part. LEMMA 1. Let (g,) be a sequence of functions in w' whose range K is Turing closed. Then there is a perfect tree T e L((gn)) such that (Vf e T)(f 0 U L(gn) -* ((Vn)(gn is Al inf andf is not Al in gn) & l(ff) = A'(K)) LEMMA 2. If (Vg)(ww t L(g)) then no w nr L(g) contains a perfect subset. PROOF. Let g E wO, Tbe a perfect subset of aU, and h O L(g, T). ChoosefE Tso that h ? T (f, T). Then f L(g). LEMMA 3. Suppose (Vg)(wU t L(g)), and fix f. Then there is a strictly increasing sequence of A-degrees df, cc < w1, such that fe do and Al(dj) = z\1(f) for all c < w1. PROOF. It is an obvious transfinite recursion on w1 by means of Lemma 1. Thanks to Lemma 2, the recursion is well-defined. The following is well-known. Received January 5, 1973. This research was partially supported by NSF GP-34091X. 79 @ 1974, Association for Symbolic Logic This content downloaded from 157.55.39.72 on Thu, 15 Sep 2016 05:42:10 UTC All use subject to http://about.jstor.org/terms