HIGHER RANDOMNESS AND LIM-SUP FORCING WITHIN AND BEYOND HYPERARITHMETIC

We develop arboreal forcing in the context of hyperarithmetical randomness theory. In any transitive model of the Kripke-Platek set theory obtained as the companion of a Spector pointclass, we show that certain kinds of Σ₁-definable lim-sup tree creature forcings have the Σ₁- continuous reading of names, the Σ₁-fusion property, and so on. In this way, we show that the shape of Cichoń’s diagram in hyperarithmetic theory is different from those in computability theory and set theory.