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This is the third of a series of papers in these Transactions on hierarchies obtained by quantifying variables of recursive predicates. In Recursive predicates and quantifiers [10] variables for natural numbers (type 0) were quantified, and in Arithmetical predicates and function quantifiers [14] also variables for one-place number-theoretic functions (type(More)
For over two millenia mathematicians have used particular examples of algorithms for determining the values of functions. The notion of "?-definability" was the first of what are now accepted as equivalent exact mathematical descriptions of the class of the functions for which algorithms exist. This article explains the notion and traces the investigation(More)
The existence of hierarchies of point sets in analysis has long been familiar from the work of Borel and Lusin. The study of the hierarchies in number theory which we consider here began with a theorem presented to the Society in 1940 and published in [12], These hierarchies have applications in foundational investigations, but we shall be concerned here(More)