Ranked Forcing and the Length of Generalized Borel Hierarchies

TL;DR

This paper extends forcing techniques to uncountable cardinals to analyze the structure and length of the generalized Borel hierarchy, constructing models with complex hierarchy configurations and determining exact complexities of well-founded trees.

Contribution

It develops a new framework for $eta$-forcing at uncountable cardinals and applies it to analyze and control the $oldsymbol{ ext{kappa}}$-Borel hierarchy's length and structure.

Findings

Constructed models with diverse $oldsymbol{ ext{kappa}}$-Borel hierarchy lengths

Established the $oldsymbol{ ext{kappa}}$-Borel complexity of well-founded trees

Generalized Steel's forcing and Stern's arguments to the uncountable setting

Abstract

We extend A. Miller's framework of $\alpha$-forcing to the case of a regular uncountable cardinal $\kappa = \kappa^{<\kappa}$ and apply it to study the structure of the $\kappa$-Borel hierarchy on subspaces of the generalized Baire space ${}^\kappa \kappa$. We isolate a class of iterations of $\alpha$-forcing and show that it satisfies a certain combinatorial property of admitting a sufficiently rich family of rank functions; this fact is then used to construct several models in which nontrivial constellations for the length of the $\kappa$-Borel hierarchy on multiple subspaces of ${}^\kappa \kappa$ are realized simultaneously. Finally, we provide a higher variant of Steel's forcing with tagged trees and generalize arguments of Stern to derive the exact $\kappa$-Borel complexity of certain classes of well-founded trees.