Definable hereditary families in the projective hierarchy
Volume 140 / 1992
Fundamenta Mathematicae 140 (1992), 183-189
DOI: 10.4064/fm-140-2-183-189
Abstract
We show that if ℱ is a hereditary family of subsets of $ω^ω$ satisfying certain definable conditions, then the $Δ_1^1$ reals are precisely the reals α such that ${β:α ∈ Δ_1^1(β)} ∉ ∈ ℱ$. This generalizes the results for measure and category. Appropriate generalization to the higher levels of the projective hierarchy is obtained under Projective Determinacy. Application of this result to the $Q_{2n+1}$-encodable reals is also shown.
