A semi-completeness theorem.
Stephen L. Bloom · Notre Dame Journal of Formal Logic · 1969
In [2] and [3] the hyperprojective hierarchy of subsets of ft (the functions from N, the natural numbers, into N) was constructed using the hyperarithmetic hierarchy as a model.As a consequence, many properties of the hyperarithmetic sets have perfect analogues in the hyperprojective sets.In this paper we consider the problem of finding a "projective analogue" for the following important fact connected with the hyperarithmetic hierarchy:The set of Gδdel numbers of the recursive well-orderings is a complete Έ\ set of natural numbers (see [5]).In [2] it was shown that the set of indices of the projective wellorderings 1 of subsets of ft is a Δ 2 set 2 and thus cannot be a "complete π 2 set" in any natural sense, as our analogy would have it.The difficulty is that in order to express the notion "there is no countable descending chain of functions such that ..." one needs only a function quantifier, not a quantifier over functions from ft into N. Thus we are led to consider the collection W* of indices of those projective linear orderings having no uncountable descending chains.In this paper we will show that W* has a semi-completeness property with respect to a subclass of the π 2 sets.Our proof will assume the existence of a projective well-ordering <* of all of ft such that ft in this ordering is order-isomorphic to the first uncountable ordinal Ω.This assumption is consistent with the usual axioms for set theory, since the existence of a Δ£ well-ordering of ft (of length Ω) follows from the Axiom of Constructibility [l].Definition.If a subset B of ft is linearly ordered by some relation