Borel $$^{*}$$ Sets in the Generalized Baire Space and Infinitary Languages

Abstract

We start by giving a survey to the theory of Borel * (κ) sets in the generalized Baire space Baire(κ) = κ κ . In particular we look at the relation of this complexity class to other complexity classes which we denote by Borel(κ), ∆ 1 1 (κ) and Σ 1 1 (κ) and the connections between Borel * (κ) sets and the infinitely deep language M κ + κ . In the end of the paper we will prove the consistency of Borel * (κ) = Σ 1 1 (κ).

Questions on generalised Baire spaces

“…It is known that both and are consistent, and that is consistent, however, the consistency of is still open; cf. for more detail. Question Is it consistent that ?…”

“…It is known that both Borel * = 1 1 and Borel * = 1 1 are consistent, and that 1 1 = Borel * is consistent, however, the consistency of 1 1 = Borel * is still open; cf. [22,32,37] for more detail. Question 3.21 (Friedman,Hyttinen,Kulikov;[22,37]) Is it consistent that 1 1 = Borel * ?…”

“…[22,32,37] for more detail. Question 3.21 (Friedman,Hyttinen,Kulikov;[22,37]) Is it consistent that 1 1 = Borel * ? Questions 3.22 to 3.27 questions refer to [49], where various sub-classes of 1 1 -sets are examined.…”

“…Cf. or the original source for more on such languages; cf. for an in‐depth exposition of model theory in this generalised setting, as well as the generalisations of the Silver and the Harrington‐Kechris‐Louveau dichotomy.…”

A generalized Borel-reducibility counterpart of Shelah’s main gap theorem

“…κ-closed forcing R(t, h) such that in any R(t, h)-generic extension ∼ = κ DLO is not a Borel * -set. The forcing in [HK12, Thm 3.1] works for every theory T that is unstable, or T non-classifiable and superstable (not only DLO, see [HK12] and [HT91]). Therefore, this claim can be generalized to:…”