A classification of projective-like hierarchies in L(R) Rachid Atmai Universit¨atWien Kurt G¨odelResearch Center W¨ahringerStraße 25 1090 Vienna
[email protected] Steve Jackson Dept. of Mathematics General Academics Building 435 1155 Union Circle 311430 University of North Texas Denton, TX 76203-5017
[email protected] October 10, 2016 Abstract We provide a characterization of projective-like hierarchies in L(R) in the context of AD by combinatorial properties of their associated Wadge ordinal. In the second chapter we concentrate on the type III case of the analysis. In the last chapter, we provide a characterization of type IV projective-like hierarchies by their associated Wadge ordinal. 1 Introduction 1.1 Outline The paper is roughly organized as follows. We first review basic definitions of the abstract theory of pointclasses. In the second section we study the type III pointclasses and the clo- sure properties of the Steel pointclass generated by a projective algebra Λ in the case where cof(o(Λ)) > !. In particular we show that Steel's conjecture is false. This leads to isolat- ing a combinatorial property of the Wadge ordinal of the projective algebra generating the 1 Steel pointclass. This combinatorial property ensures closure of the Steel pointclass under dis- junctions. Therefore it is still possible to have κ regular, where κ is the Wadge ordinal for a projective algebra Λ generating the next Steel pointclass and yet the Steel pointclass is not closed under disjunction. In the third section, we look at the type IV projective-like hierarchies, that is the pointclass which starts the new hierarchy is closed under quantifiers.