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Iteration theorems for subversions of forcing classes

Gunter Fuchs, Corey Bacal Switzer

TL;DR

This work develops two parallel frameworks for iterating forcing notions tied to Jensen's subproper and subcomplete classes: revised countable support (RCS) iterations and Miyamoto's nice iterations. It proves that both approaches yield iterable forcing classes that preserve $\omega_1$, with subcomplete iterations additionally not adding reals, and shows preservation of Souslin trees and $\omega_1$-trees under various iteration schemes. A key innovation is the introduction of $\infty$-subcomplete and $\infty$-subproper$ notions, which drop the hull condition yet retain iterability, enabling broader iteration theorems equated across RCS and nice-iteration formalisms. The paper culminates in applications to models of the subcomplete forcing axiom $SCFA$, including $SCFA+\neg CH$ and related cardinal characteristics, demonstrating rich interactions between forcing axioms, tree properties, and the continuum. Overall, the results offer robust, parallel frameworks for iterating subproper/subcomplete forcing and yield new models where strong forcing axioms co-exist with varied continuum configurations.

Abstract

We prove various iteration theorems for forcing classes related to subproper and subcomplete forcing, introduced by Jensen. In the first part, we use revised countable support iterations, and show that 1) the class of subproper, ${}^ωω$-bounding forcing notions, 2) the class of subproper, $T$-preserving forcing notions (where $T$ is a fixed Souslin tree) and 3) the class of subproper, $[T]$-preserving forcing notions (where $T$ is an $ω_1$-tree) are iterable with revised countable support. In the second part, we adopt Miyamoto's theory of nice iterations, rather than revised countable support. We show that this approach allows us to drop a technical condition in the definitions of subcompleteness and subproperness, still resulting in forcing classes that are iterable in this way, preserve $ω_1$, and, in the case of subcompleteness, don't add reals. Further, we show that the analogs of the iteration theorems proved in the first part for RCS iterations hold for nice iterations as well.

Iteration theorems for subversions of forcing classes

TL;DR

This work develops two parallel frameworks for iterating forcing notions tied to Jensen's subproper and subcomplete classes: revised countable support (RCS) iterations and Miyamoto's nice iterations. It proves that both approaches yield iterable forcing classes that preserve , with subcomplete iterations additionally not adding reals, and shows preservation of Souslin trees and -trees under various iteration schemes. A key innovation is the introduction of -subcomplete and -subproperSCFASCFA+\neg CH$ and related cardinal characteristics, demonstrating rich interactions between forcing axioms, tree properties, and the continuum. Overall, the results offer robust, parallel frameworks for iterating subproper/subcomplete forcing and yield new models where strong forcing axioms co-exist with varied continuum configurations.

Abstract

We prove various iteration theorems for forcing classes related to subproper and subcomplete forcing, introduced by Jensen. In the first part, we use revised countable support iterations, and show that 1) the class of subproper, -bounding forcing notions, 2) the class of subproper, -preserving forcing notions (where is a fixed Souslin tree) and 3) the class of subproper, -preserving forcing notions (where is an -tree) are iterable with revised countable support. In the second part, we adopt Miyamoto's theory of nice iterations, rather than revised countable support. We show that this approach allows us to drop a technical condition in the definitions of subcompleteness and subproperness, still resulting in forcing classes that are iterable in this way, preserve , and, in the case of subcompleteness, don't add reals. Further, we show that the analogs of the iteration theorems proved in the first part for RCS iterations hold for nice iterations as well.

Paper Structure

This paper contains 16 sections, 35 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. RCS iterations
  3. The subproperness extension lemma
  4. RCS and nicely subproper iterations
  5. Iterating subproper Souslin tree preserving forcing
  6. Nicely subproper iterations of $[T]$-preserving forcing
  7. RCS and nicely subproper iterations of ${}^\omega\omega$-bounding forcing
  8. Nice Iterations
  9. Subcompleteness and $\infty$-subcompleteness
  10. The theory of nice iterations
  11. Nice iterations of $\infty$-subcomplete forcing
  12. Nice iterations of $\infty$-subproper forcing
  13. Preserving Properties of Trees
  14. Nice iterations of $\infty$-subproper ${}^\omega\omega$-bounding forcing
  15. Applications
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assuming the consistency of a supercompact cardinal the following are consistent with $\mathsf{SCFA}\xspace + \neg \mathsf{CH}\xspace$.

Theorems & Definitions (76)

  • Theorem 1.1
  • Definition 2.1
  • proof
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5: Subproperness Extension Lemma
  • proof
  • Definition 2.6
  • Theorem 2.9: Jensen
  • Definition 2.10: after Jensen
  • ...and 66 more