Ranked Forcing and the Length of Generalized Borel Hierarchies

Abstract: We extend A. Miller's framework of $α$-forcing to the case of a regular uncountable cardinal $κ= κ<sup>{&lt;κ}$ and apply it to study the structure of the $κ$-Borel hierarchy on subspaces of the generalized Baire space ${}<sup>κκ$. We isolate a class of iterations of $α$-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 $κ$-Borel hierarchy on multiple subspaces of ${}<sup>κκ$ 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 $κ$-Borel complexity of certain classes of well-founded trees.