On Mathias generic sets

TL;DR

This paper investigates the hierarchy and complexity of generics in computable Mathias forcing, revealing their structural properties, degrees, and relationships with Cohen generics and bi-immune sets.

Contribution

It establishes the hierarchy of n-generics in Mathias forcing, analyzes their complexity, and explores their degrees and computational capabilities.

Findings

n-generics form a strict hierarchy similar to Cohen forcing

n-generics with n ≥ 3 satisfy a specific jump property

Such n-generics have generalized high degrees and can compute Cohen n-generics with bi-immune sets

Abstract

We present some results about generics for computable Mathias forcing. The $n$-generics and weak $n$-generics in this setting form a strict hierarchy as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if $G$ is any $n$-generic with $n \geq 3$ then it satisfies the jump property $G^{(n-1)} = G' \oplus \emptyset^{(n)}$. We prove that every such $G$ has generalized high degree, and so cannot have even Cohen 1-generic degree. On the other hand, we show that $G$, together with any bi-immune set $A \leq_T \emptyset^{(n-1)}$, computes a Cohen $n$-generic set.