Several problems that pertain to certain arithmetically well-behaved countable subsemirings of Λ, the semiring of isols, are discussed. This is relevant to the present volume memorializing the late John Myhill, in that Myhill was an early co-developer of the theory of Λ.

In this paper we discuss the following contributions to recursion theory made by John Myhill: two sets are recursively isomorphic iff they are one-one equivalent; two sets are recursively isomorphic iff they are recursively equivalent and their complements are also recursively equivalent; every two creative sets are recursively isomorphic; the recursive analogue of the Cantor–Bernstein theorem; the notion of a combinatorial function and its use in the theory of recursive equivalence types.

We construct a recursive ultrapower F/U such that F/U is a tame 1-model in the sense of [6, §3] and FU is existentially incomplete in the models of II2 arithmetic. This enables us to answer in the negative a question about closure with respect to recursive fibers of certain special semirings Γ of isols termed tame models by Barback. Erik Ellentuck had conjuctured that all such semirings enjoy the closure property in question. Our result is that while many do, some (...) do not. (shrink)

We solve a question of McLaughlin by showing that if A is a regressive co-simple isol, there is a co-simple regressive isol B such that the intersection type of A and B is trivial. The proof is a nonuniform 0 priority argument that can be viewed as the execution of a single strategy from a 0-argument. We establish some limit on the properties of such pairs by showing that if AxB has low degree, then the intersection type of A and (...) B cannot be trivial. (shrink)