Notions of indifference for genericity: union sets and subsequence sets

摘要

A set I is said to be a universal indifferent set for 1-genericity if for every 1-generic G and for all X subset of I, G Delta X is also 1-generic. Miller (2013, The Journal of Symbolic Logic, 78, 113-138) 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.