Limiting recursion

A class of problems is called decidable if there is an algorithm which will give the answer to any problem of the class after a finite length of time. The purpose of this paper is to discuss the classes of problems that can be solved by infinitely long decision procedures in the following sense: An algorithm is given which, for any problem of the class, generates an infinitely long sequence of guesses. The problem will be said to be solved in the limit if, after some finite point in the sequence, all the guesses are correct and the same (in case there is more than one correct answer). Functions, sets, and functionals which are decidable by such infinite algorithms will be called limiting recursive . These, together with classes of objects which can be identified in the limit , are the subjects of this report. Without qualification, set will mean set of numbers; function will mean number-theoretic function of 1 variable, possibly partial; functionals will take numerical values and have any number of numerical and/or function variables, the latter ranging solely over total functions of 1 variable. Thus a function is a special case of a functional, x will invariably stand for a numerical variable; φ for a function variable; g for a guess (a number); n for the numerical variable which indexes the guesses, referred to as the time .