Randomness below complete theories of arithmetic

TL;DR

This paper demonstrates that degrees containing complete extensions of arithmetic possess the random join property, allowing them to be expressed as the supremum of any random real they compute along with another random real.

Contribution

It establishes that such degrees have the random join property across various reducibilities, extending understanding of their structure in computability theory.

Findings

Degrees with complete extensions of arithmetic have the random join property.

The property holds for truth-table and weak truth-table reducibilities.

These degrees can be represented as the supremum of two random reals.

Abstract

We show that degrees containing a complete extensions of arithmetic have the random join property: they are the supremum of any random real they compute, with another random real. The same is true for the truth-table and weak truth-table reducibilities.