An infinite product of isols
J. C. E. Dekker · Illinois Journal of Mathematics · 1963
IntroductionLet us denote the set of all numbers (i.e., nonnegative integers) by e, and the class of all sets (i.e., subcollections of e) by V.If f(x) is a function from a subset of e into , we write f and of for its domain and its range re- spectively.The set a is recursively equivalent to the set/3 (written if there is a partial recursive one-to-one function p () such that a p and p (a) /.This relation between sets is reflexive, symmetric, and transi- rive.The class of all sets z such that z a is denoted by Req c.Using the relation one can extend the system [e, -k, consisting of the set e and the binary operations of ordinary addition and multiplication to the system [A, +, of all isols.The method by which this extension can be obtained isThere is a subcollection of A, called the collection of all regressive isols, which plays a special role in the present paper.We shall therefore recall its definition and some of its properties.A function t, from e into e is re- gressive if it is one-to-one and there is a partial recursive function p (x) such that pt p, p (to) to, and p (t+) t, for every n.A set is regressive if it is finite or the range of some regressive function.Every set which is recursively equivalent to a regressive set is itself regressive; also, every re- gressive set is recursively enumerable or immune.An isol is regressive if it contains at least one regressive set, or equivalently, if it contains only re- gressive sets.The collection of all regressive isols is denoted by A. Both A and A have cardinality c.Let t and t* be one-to-one functions from e into e.Then t is recursively equivalent to t*, (written t --t'n) if there is a partial reeursive one-to-one function p (z) such that pt p and p (t) tn* for every n.This relation between one-to-one functions from e into e is also reflexive, symmetric, and transitive.The basic property of regressive functions is as follows.Let t and t*n be one-to-one functions from with respective ranges r and r*.If t and t* are regressive functions, t t* * This enables us to ssocite with every infinite regressive isol T denumerable fmily of functions, namely the fmily of M1 regressive functions rnging over sets in T. It cn be shown that if T is a regressive isol, so is 2 r.In [4] the sum of n infinite series of finite isols (i.e., ordinary numbers) ws defined, provided the summation is performed with respect to n infinite