Canonical immunity and genericity
Achilles A. Beros, Konstantinos A. Beros
TL;DR
This paper explores the concept of canonical immunity linked to Mathias genericity, contrasting it with Cohen genericity, and establishes key relationships between these notions in computability theory.
Contribution
It introduces the notion of canonical immunity associated with Mathias genericity and clarifies its distinction from Cohen genericity, providing new insights into their interactions.
Findings
Every Mathias generic is canonically immune.
No Cohen 2-generic computes a canonically immune set.
Abstract
Whereas the usual notions of immunity -- e.g., immunity, hyperimmunity, etc. -- are associated with Cohen genericity, canonical immunity, as introduced by Beros, Khan and Kjos-Hanssen, is associated instead with Mathias genericity. Specifically, every Mathias generic is canonically immune and no Cohen 2-generic computes a canonically immune set.
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