# The basic theory of partial α-recursive operators

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## Summary

*In this paper, we investigate the theory of partial α-recursive operators and functionals, α an admissible ordinal, which are defined in terms of α-enumeration reducibility. The theory bifurcates into the study of weak operators and functionate, and of operators and functionate proper. The status of the representative theorems of the classical theory (when α=ω) is examined relative to both kinds of operators and functionals. Especial attention is given to the difficulties, when such exist, encountered in generalizing a classical result, whether simple or profound, to level α. In the course of the investigation we are led to consider briefly topics such as the structure theory of completely recursively enumerable classes of α-recursively enumerable sets. This is natural since this theory bears on the properties of effective operations at level α. The paper provides the framework for the further investigation of this and allied topics*.

### Keywords

Classical Theory Classical Result Basic Theory Structure Theory Weak Operator## Preview

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