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Ranked Forcing and the Length of Generalized Borel Hierarchies

Nick Chapman

Abstract

We extend A. Miller's framework of $α$-forcing to the case of a regular uncountable cardinal $κ= κ^{<κ}$ and apply it to study the structure of the $κ$-Borel hierarchy on subspaces of the generalized Baire space ${}^κκ$. 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 ${}^κκ$ 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.

Ranked Forcing and the Length of Generalized Borel Hierarchies

Abstract

We extend A. Miller's framework of -forcing to the case of a regular uncountable cardinal and apply it to study the structure of the -Borel hierarchy on subspaces of the generalized Baire space . We isolate a class of iterations of -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 -Borel hierarchy on multiple subspaces of 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 -Borel complexity of certain classes of well-founded trees.
Paper Structure (20 sections, 39 theorems, 80 equations)

This paper contains 20 sections, 39 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries and notation
  4. Topology
  5. Trees
  6. Forcing
  7. Borel sets
  8. Coding Borel sets
  9. Order
  10. Ranked forcing
  11. $\alpha$-forcing
  12. Regularity and iterability
  13. Preserving the generic $\Pi$-set
  14. Iterating $\alpha$-forcing
  15. Iterations with ground model spaces
  16. ...and 5 more sections

Key Result

Theorem 1

Suppose $X \subseteq \tensor[^{\kappa}]{{\kappa}}{}$ has size $|X| > \kappa$ and $1 < \alpha < \omega$ is a finite ordinal. Then there exists a ${<}\kappa$-closed, $\kappa^+$-c.c. forcing extension $V[G]$ such that

Theorems & Definitions (108)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Remark 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 4.1
  • Theorem 4.2: miller_descriptive_1995
  • ...and 98 more