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Strongly increasing sequences

Paul B. Larson, Chris Lambie-Hanson

TL;DR

The paper studies the existence and limits of strongly increasing sequences of functions from $\omega$ to ordinals, linking these sequences to variants of Chang's Conjecture and inner-model questions. It constructs a very strongly increasing $\omega_2$-sequence in a ZFC model obtained by forcing over a background model with $\mathsf{AD}^+$ and $V=L(\mathbb{R})$, via a variation of Woodin's $\mathbb{P}_{\mathrm{max}}$. A PCF-theoretic bound rules out longer sequences (in particular, no such sequence of length $\omega_4$), clarifying the maximal possible length under these methods. The approach combines a $\mathbb{P}_{\mathrm{max}}$-like forcing with Hechler forcing and club-shooting to achieve $2^{\aleph_0}=\aleph_2$ and to realize the very strongly increasing sequence, while outlining open questions about the possibility of a length $\omega_3$ sequence and the precise consistency strength involved.

Abstract

Using a variation of Woodin's $\mathbb{P}_{\mathrm{max}}$ forcing, we force over a model of the Axiom of Determinacy to produce a model of ZFC containing a very strongly increasing sequence of length $ω_{2}$ consisting of functions from $ω$ to $ω$. We also show that there can be no such sequence of length $ω_{4}$.

Strongly increasing sequences

TL;DR

The paper studies the existence and limits of strongly increasing sequences of functions from to ordinals, linking these sequences to variants of Chang's Conjecture and inner-model questions. It constructs a very strongly increasing -sequence in a ZFC model obtained by forcing over a background model with and , via a variation of Woodin's . A PCF-theoretic bound rules out longer sequences (in particular, no such sequence of length ), clarifying the maximal possible length under these methods. The approach combines a -like forcing with Hechler forcing and club-shooting to achieve and to realize the very strongly increasing sequence, while outlining open questions about the possibility of a length sequence and the precise consistency strength involved.

Abstract

Using a variation of Woodin's forcing, we force over a model of the Axiom of Determinacy to produce a model of ZFC containing a very strongly increasing sequence of length consisting of functions from to . We also show that there can be no such sequence of length .

Paper Structure

This paper contains 3 sections, 9 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Chang's Conjecture and strongly increasing sequences
  3. Consistency via a $\mathbb{P}_{\mathrm{max}}$ variation

Key Result

Theorem 4

Suppose that $V =L(A, \mathbb{R})$, for some $A \subseteq \omega^{\omega}$, and that $\mathsf{AD}^{+}$ holds. Then there is a forcing extension in which ZFC holds and there exists a very strongly increasing $\omega_{2}$-sequence of functions from $\omega$ to $\omega$.

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Remark 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • proof
  • Claim 7
  • proof
  • Lemma 10
  • ...and 15 more