Higher-order indecomposable isols
Alfred B. Manaster · Transactions of the American Mathematical Society · 1966
An effective analogue of the theory of cardinal numbers was created about ten years ago by J. C. E. Dekker.(See .)In it the only sets considered are subsets of the natural numbers and the only functions considered are 1-1 partial recursive functions.In the classical theory a cardinal number may be considered to be an equivalence class containing all sets for which there is a 1-1 function mapping the set onto a given set.The Dekker analogue of a cardinal number is an equivalence class on the set of all subsets of E (E={0, 1, 2,...}) containing all sets a for which there is a 1-1 partial recursive function/such that the domain of/includes a and the image of a under/is a given subset of E. These equivalence classes are called recursive equivalence types, or RETs.The collection of all RETs is denoted £2.The RET to which a set, a, belongs is denoted .Addition is defined on the RETs in the following manner.If aç£ and ß^E, a and ß are called recursively separated if there exist disjoint recursively enumerable (RE) sets o) and 6 such that aÇw and ߣ 6.Let A and B be RETs.The sum of A and B is defined to be the RET represented by a u ß where A = (a}, ¿?=, and a and ß are recursively separated.We define A ÚB for RETs A and B if there is an RET C such that A + C=B.The ¿ relation is a partial ordering of the RETs.If A ¿ B we say A is a predecessor of B. An RET is called an isol if it satisfies the additive cancellation law.The collection of all isols is denoted A. Thus A e A if and only if for all RETs B and C, A + B = A + C implies B=C.A subset, a, of E is called isolated if there is no 1-1 partial recursive function / whose domain includes a and such that f(a) §¡ a. e A if and only if a is isolated.A subset of E contains no infinite RE subset if and only if it is isolated.Thus A is the collection of equivalence classes of sets which do not have infinite RE subsets.A is closed under addition and predecessor.The assertions of this paragraph are proved in [4].Dekker-Myhill [4, p. 114] define an ideal in A as a subsystem of A closed under addition and predecessor.A sequence of ideals will be defined here in order to discuss the results of this paper.Variables X, Y, Z, V, W range over A.