As to the misconceptions: In the first place, the existence of "undecidable propositions" or "unsolvable problems" has only remote connections with the failure of excluded middle. More precisely, from the fact that a certain problem is unsolvable, one cannot infer that the affirmative and negative answers to that problem are both incorrect. Both Gödel's and Church's theorems were originally proved for systems with the excluded middle, i.e. for systems in which 'p or not p' is provable for every proposition 'p'; (...) though in the light of Gödel's theorem there are propositions 'p' such that neither 'p' nor 'not p' is provable in those systems. Careful distinction should be made between the following locutions. (shrink)

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. Functions, sets, and functionals which are decidable by such infinite algorithms will be calledlimiting recursive. These, together with classes of objects which can beidentified in the limit, are the subjects of this report.Without qualification,setwill mean set of numbers;functionwill mean number-theoretic function of 1 variable, possibly partial;functionalswill 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,xwill invariably stand for a numerical variable;φfor a function variable;gfor a guess ;nfor the numerical variable which indexes the guesses, referred to as thetime. (shrink)