Diophantine correct non-standard models in the isols

Dedekind introduced a non-standard definition of finiteness as follows. A set a is Dedekind finite if there is no 1-1 function f, defined on a, which maps a onto a proper subset of a. (In the absence of the axiom of choice, there can be Dedekind finite sets which are not finite in the usual inductive sense.) Dekker [1] introduced a relativised notion of Dedekind finiteness, where sets a are subsets of the set E -{O, 1, 2, .. . } of non-negative integers, and 1-1 functions f are 1-1 partial recursive functions. A subset a of E is called isolated if there is no 1-1 partial recursive function f, defined on a, such that f(a) ; a. This gave rise to an effective analogue of the theory of cardinals of Dedekind finite sets, the theory of isols. Sets a, 18 c E are called recursively equivalent if there is a 1-1 partial recursive function p, at least defined on a, such that p(a) 1 8. The equivalence class of an isolated set a is an isol, and A is the set of isols. Operations of addition and multiplication for isols are given by