Notions of indifference for genericity: Union and subsequence sets

TL;DR

This paper explores the concept of universal indifferent sets for 1-genericity, proving their non-existence under certain variants, but identifies a non-computable subsequence set for weak-1-genericity.

Contribution

It introduces two variants of universal indifferent sets for 1-genericity and proves their non-existence, while also demonstrating a non-computable subsequence set for weak-1-genericity.

Findings

No non-trivial universal sets for 1-genericity under the variants.

Existence of a non-computable subsequence set for weak-1-genericity.

Abstract

A set $I$ is said to be a universal indifferent set for $1$-genericity if for every $1$-generic $G$ and for all $X \subseteq I$, $G \Delta X$ is also $1$-generic. Miller showed that there is no infinite universal indifferent set for $1$-genericity. We introduce two variants (union and subsequence sets for $1$-genericity) of the notion of universal indifference and prove that there are no non-trivial universal sets for $1$-genericity with respect to these notions. In contrast, we show that there is a non-computable subsequence set for weak-$1$-genericity.