Hyperarithmetical Index Sets in Recursion Theory

Hyperarithmetical Index Sets in Recursion Theory

We define a family of properties on hyperhypersimple sets and show that they yield index sets at each level of the hyperarithmetical hierarchy. An extension yields a nj-complete index set. We also classify the index set of quasimaximal sets, of coinfinite r.e. sets not having an atomless superset, and of r.e. sets major in a fixed nonrecursive r.e. set. 0. Introduction. The present paper deals ...

Hyperarithmetical Sets

1. Preamble: Kleene [1943], Post [1944] and Mostowski [1947] . . . . . . . . . 2 1A. Post’s degrees of unsolvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1B. Kleene’s arithmetical hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1C. Kleene [1943] vs. Post [1944] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1D. Mostowski [1...

Intrinsically Hyperarithmetical Sets

The main result proved in the paper is that on every recursive structure the intrinsically hyperarithmetical sets coincide with the relatively intrinsically hyperarithmetical sets. As a side eeect of the proof an eeective version of the Kueker's theorem on deenability by means of innnitary formulas is obtained. 1. Introduction One of the main achievements of the classical recursion theory is th...

Rudimentary recursion and provident sets

This paper, a contribution to “micro set theory”, is the study promised by the first author in [M4], but improved and extended by work of the second. We use the rudimentarily recursive (set theoretic) functions and the slightly larger collection of gentle functions to develop the theory of provident sets, which are transitive models of PROVI, a very weak subsystem of KP which nevertheless suppo...

Ordinal Recursion Theory