Higher Randomness and Forcing with Closed Sets

Benoit Monin (1)

1 LACL - Laboratoire d'Algorithmique Complexité et Logique

Kechris showed in [9] that there exists a largest Π11 set of measure 0. Due to its universal nature, it was conjectured by many that this nullset has a high Borel rank. In this paper, we refute this conjecture and show that this nullset is merely Σ03. Together with a result of Liang Yu, our result also implies that the exact Borel complexity of this set is Σ03. To do this proof, we develop the machinery of effective randomness and effective Solovay genericity, investigating the connections between those notions and effective domination properties.

Mots clés: Effective descriptive set theory; Higher computability; Effective randomness; Genericity

Domaines: Mathématiques [math]; Informatique [cs]

Benoit Monin. Higher Randomness and Forcing with Closed Sets. Theory of Computing Systems, 2016, ⟨10.1007/s00224-016-9681-5⟩. ⟨hal-01397294⟩