CO‐Simple Higher‐Order Indecomposable Isols

Recursive equivalence types have been studied as a recursive analogue of the notion of cardinal number since their introduction by DEKKER [4]. Some of the earliest notions of Dedekind finite RETs were provided by cohesive sets (in modern terminology) and retraceable sets ([3] and [5]). The cardinal-theoretic properties of these sets as RETs are easily and accurately described by the terms indecomposable and highly decomposablr respectively. These notions were generalized in [12] where a hierarchy of levels of indecomposable and highly-decomposable notions was defined. It was seen that the hierarchy included members a t levels corresponding to every countable ordinal and. conversely, that every Dedekind-finite RET did f i t into some countable level of the hierarchy. A further restriction on the notion of effectiveness for analogues of cardinal numbers is provided by requiring the compfements of the sets considered to be recursively enumerable ([6] and [9]). Recursively enumerable sets whose complements are recursively Dedekind finite are just the so-called simple sets of POST [14]. The purpose of this paper is to provide examples of co-simple RETs which are indecomposable and highly decomposable a t levels corresponding to every recursive ordinal. A co-simple isol which is highly decomposable a t some non-recursive level , 9 is also given. DEKKER and MYHILL [6] also mentioned the ideal of co-hypersimple RETs. All of the examples here are co-hyperhypersimple. ELLENTUCK has raised the question of the further restriction of RETs to sets satisfying various conditions on their degrees of unsolvability. ,411 of our examples may be in any high degree, i.e., a degree whose jump is 0“. (See