COMPUTABILITY IN UNCOUNTABLE BINARY TREES

Computability, while usually performed within the context of omega, may be extended to larger ordinals by means of alpha-recursion. In this article, we concentrate on the particular case of omega(1)-recursion, and study the differences in the behavior of Pi(0)(1)-classes between this case and the standard one. Of particular interest are the Pi(0)(1)-classes corresponding to computable trees of countable width. Classically, it is well-known that the analog to Konig's Lemma-"every tree of countable width and uncountable height has an uncountable branch"-fails; we demonstrate that not only does it fail effectively, but also that the failure is as drastic as possible. This is proven by showing that the omega(1)-Turing degrees of even isolated paths in computable trees of countable width are cofinal in the Delta(1)(1) omega(1)-Turing degrees. Finally, we consider questions of nonisolated paths, and demonstrate that the degrees realizable as isolated paths and the degrees realizable as nonisolated ones are very distinct; in particular, we show that there exists a computable tree of countable width so that every branch can only be omega(1)-Turing equivalent to branches of trees with N-2-many branches.