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An Iteration Theorem for $ω_1$-preserving Forcings

Andreas Lietz

Abstract

We prove an iteration theorem which guarantees for a wide class of nice iterations of $ω_1$-preserving forcings that $ω_1$ is not collapse, at the price of needing large cardinals to burn as fuel. More precisely, we show that a nice iteration of $ω_1$-preserving forcings which force SRP at successor steps and preserves old stationary sets does not collapse $ω_1$.

An Iteration Theorem for $ω_1$-preserving Forcings

Abstract

We prove an iteration theorem which guarantees for a wide class of nice iterations of -preserving forcings that is not collapse, at the price of needing large cardinals to burn as fuel. More precisely, we show that a nice iteration of -preserving forcings which force SRP at successor steps and preserves old stationary sets does not collapse .
Paper Structure (11 sections, 29 theorems, 82 equations)

This paper contains 11 sections, 29 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Iterations of c.c.c. Forcings
  3. Iterations of Proper Forcings
  4. Iterations of Semiproper Forcings
  5. Iterations of Stationary-Set-Preserving Forcing
  6. Iterations of $\omega_1$-Preserving Forcings
  7. Notation
  8. $\diamondsuit(\mathbb B)$ and $\diamondsuit^{+}(\mathbb{B})$
  9. Miyamoto's theory of nice iterations
  10. The Iteration Theorem
  11. $f$-Proper and $f$-Semiproper Forcings

Key Result

Theorem 1.1

Suppose $\langle \mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid\alpha\leq\gamma, \beta<\gamma\rangle$ is a finite support iteration of c.c.c. forcings. Then $\mathbb{P}_\gamma$ is c.c.c..

Theorems & Definitions (76)

  • Theorem 1.1: Solovay-Tennenbaum, solten
  • Theorem 1.2: Solovay-Tennenbaum, solten
  • Definition 1.3: Shelah,shelahbook
  • Theorem 1.4: Shelah,shelahbook
  • Theorem 1.5: Abraham-Shelah, abrshelahtrees
  • Definition 1.6: Shelah, shelahbook
  • Theorem 1.7: Shelah
  • Theorem 1.8: Shelah, see nssat for a proof
  • Theorem 1.9: Steel, Jensen-Steel JensenSteelKWithoutAMeasurable
  • Theorem 1.10: Shelah shelahbook
  • ...and 66 more