S. C. Kleene

Hierarchies of number-theoretic predicates

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 with the exploration of their structure (using classical logic). We shall survey the previous results from the beginning, and conclude with a few new ones. We have endeavored to make the exposition complete enough so that the layman in this field can get the gist of the arguments without consulting the references.

Algorithms in various contexts

A more complicated situation is that the device may give inputs to a computation zero, one or more (but, of course, only finitely many) times, the interventions of the device being triggered by computational situations, but the device being oblivious of them in its input. We can represent this by a modification of a Turing oracle-machine (about which I will say more presently, in responding to Tseitin), which has

The Writing of Introduction to Metamathematics

It was suggested that I talk here on the writing of my book Introduction to Metamathematics (IM).