Ranked Forcing and the Length of Generalized Borel Hierarchies

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.