Weakly 2-randoms and 1-generics in Scott sets

TL;DR

This paper explores the existence of weakly 2-random and 1-generic degrees within Scott sets and models of WWKL, demonstrating their implications for Turing incomparability and logical sentences in degree structures.

Contribution

It establishes the presence of weakly 2-random and 1-generic degrees in Scott sets and models of WWKL, and shows their impact on Turing incomparability and logical properties of degree structures.

Findings

Existence of weakly 2-random degrees in Scott sets

Existence of 1-generic degrees in Scott sets

Turing incomparability among non-computable elements

Abstract

Let $S$ be a Scott set, or even an $\omega$-model of $\mathsf{WWKL}$. Then for each $A\in S$, either there is $X \in S$ that is weakly 2-random relative to $A$, or there is $X\in S$ that is 1-generic relative to $A$. It follows that if $A_1,\dots, A_n \in S$ are non-computable, there is $X \in S$ such that each $A_i$ is Turing incomparable with $X$, answering a question of Ku\v{c}era and Slaman. More generally, any $\forall\exists$ sentence in the language of partial orders that holds in $\mathcal D$ also holds in $\mathcal D_S$, where $\mathcal D_S$ is the partial order of Turing degrees of elements of $S$.