The Perfect Set Theorem and Definable Wellorderings of the Continuum

Let r be a collection of relations on the reals and let M be a set of reals. We call M a perfect set basis for r if every set in r with parameters from M which is not totally included in M contains a perfect subset with code in M. A simple elementary proof is given of the following result (assuming mild regularity conditions on r and M): If M is a perfect set basis for r, the field of every wellordering in r is contained in M. An immediate corollary is Mansfield's Theorem that the existence of a 12 wellordering of the reals implies that every real is constructible. Other applications and extensions of the main result are also given. ?1. Preliminaries. Let co = {0, 1, 2, ... } be the set of natural numbers and q = cl) the set of all functions from c) to c) or (for simplicity) reals. We study subsets of the product spaces a = X1 x X2 x ... x Xk, where Xi is co or S. We call such subsets pointsets. Sometimes we think of them as relations and we write interchangeably x E A A (x). A pointclass is a class of pointsets, usually in all product spaces. We shall be concerned primarily in this paper with the analytical pointclasses In', 171,, AI and their corresponding projective pointclasses 2l, HI, J4. For information about them we refer to [8], [11] and [12]. For a pointclass r, Determinacy (r) abbreviates the statement: Every set of reals in r is determined. Projective determinacy is the hypothesis that every projective set is determined. For information about games, determinacy, etc., the reader can consult [1], [8] and [9]. We shall make considerable use of perfect sets of reals in the following. To avoid unnecessary repetition we assume that a perfect set is always nonempty. If P c q is perfect then a code of P is a real coding in any reasonable fashion the tree associated with P i.e. the set of all finite sequences from c) which are initial segments of elements of P. We shall also talk frequently about continuous functions mapping closed subsets of R into S. Any such function can be completely described by a countable amount of information (e. g. its values at a reasonable countable dense subset of its domain), which in turn can be coded by a real called a code of the given continuous function. Sometimes it will be convenient to work with the subspace 20 of q consisting of all binary reals. In this case, it is well known that for every perfect set P c 20 Received November 15, 1976. 'Research partially supported by NSF Grant MPS75-07562.