Relatively computably enumerable reals

TL;DR

This paper investigates the properties of relatively computably enumerable reals, showing that nonempty Pi^0_1 classes contain reals not relatively c.e., and that 1-generic reals are relatively simple and above.

Contribution

It establishes new results about the existence of non-relatively c.e. reals in Pi^0_1 classes and characterizes 1-generic reals as relatively simple and above.

Findings

Nonempty Pi^0_1 classes contain reals not relatively c.e.

Every 1-generic real is relatively simple and above.

Relatively c.e. reals have specific structural properties.

Abstract

A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and Y does not compute X. A real X is relatively simple and above if there is a real Y <_T X such that X is c.e.(Y) and there is no infinite subset Z of the complement of X such that Z is c.e.(Y). We prove that every nonempty Pi^0_1 class contains a member which is not relatively c.e. and that every 1-generic real is relatively simple and above.