Cohesive Sets: Countable and Uncountable

We show that many uncountable admissible ordinals (including some cardinals) as well as all countable admissible ordinals have cohesive subsets. Exactly which cardinals have cohesive subsets, however, is shown to depend on set-theoretic assumptions such as V=L or a large cardinal axiom. The study of recursion theory on the ordinals was initiated by Takeuti and then generalized by several others to all admissible ordinals. The analogy with ordinary recursion theory has been quite striking and particularly so for the theory of degrees of unsolvability. Many major theorems, especially ones about recursively enumerable degrees have been successfully generalized to all admissible ordinals. Thus for example, Sacks and Simpson [3] have introduced the finite injury priority argument into a-recursion theory to construct two incomparable a-r.e. degrees. Indeed even an infinite injury argument has been successfully adapted to this general setting to prove that the a-r.e. degrees are dense for every admissible a [5]. In general it seems fair to say that although the proofs are often somewhat different and usually more complicated than those in ordinary recursion theory, the theorems about degrees (at least the r.e. ones) seem to carry over. The situation changes drastically when one turns from degrees to sets even if one restricts one's attention to the recursively enumerable sets. Of course the phenomena of nonregularity poses many interesting problems along these lines [4], [7] but a more striking example is provided by the notion of maximal set. (An a-r.e. set is maximal if and only if its complement is a-infinite but cannot be split into two a-infinite pieces by an a-r.e. set.) It is a well-known theorem of Friedberg that such sets exist in ordinary recursion theory [2]. On the other hand, Lerman and Simpson [1] have shown that there are no maximal r.e. sets for any uncountable admissible. Finally, if we drop the requirement of recursive enumerability, the situation becomes even further removed from that of ordinary recursion Received by the editors December 11, 1972. AMS (MOS) subject classifications (1970). Primary 02F27; Secondary 02K05.