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Increasing the second uniform indiscernible by strongly ssp forcing

Ben De Bondt, Boban Velickovic

Abstract

We introduce a new and natural stationary set preserving forcing $\mathbb P^{c-c}(λ,μ)$ that (under $\mathsf{NS}_{ω_1}$ precipitous + existence of $H_θ^#$ for a sufficiently large regular $θ$) increases the second uniform indiscernible $\mathbf{u}_2$ beyond some given ordinal $λ$. The forcing $\mathbb P^{c-c}$ shares this property with forcings defined in [2] and [9]. As a main tool we use certain natural open two player games which are of independent interest, viz. the capturing games $\mathbf{G}_M^{cap}(X)$ and the catching-capturing games $\mathbf{G}_M^{c-c}(X)$. In particular, these games are used to isolate a special family of countable elementary submodels $M \prec H_θ$ that occur as side conditions in $\mathbb P^{c-c}$ and thus allow to control the forcing in a strong way.

Increasing the second uniform indiscernible by strongly ssp forcing

Abstract

We introduce a new and natural stationary set preserving forcing that (under precipitous + existence of for a sufficiently large regular ) increases the second uniform indiscernible beyond some given ordinal . The forcing shares this property with forcings defined in [2] and [9]. As a main tool we use certain natural open two player games which are of independent interest, viz. the capturing games and the catching-capturing games . In particular, these games are used to isolate a special family of countable elementary submodels that occur as side conditions in and thus allow to control the forcing in a strong way.
Paper Structure (7 sections, 14 theorems, 37 equations)

This paper contains 7 sections, 14 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Iterable structures
  4. The second uniform indiscernible
  5. The strongly ssp property
  6. Games: catching and capturing
  7. The forcing $\mathbb{P}^{c\text{-}c}(\lambda,\mu)$

Key Result

Lemma 2.3

Suppose $\mathscr{H}$ is a transitive $\mathsf{ZFC}^\bullet$-model with $\omega_1 \subseteq \mathscr{H}$ and $\theta \in \mathscr{H}$ is an ordinal such that If $M\prec \mathscr{H}$ is countable and contains $\theta,$ then the transitive collapse of $M \cap H^\mathscr{H}_\theta$ is an iterable $\mathsf{ZFC}^\bullet$-model.

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6: woodin
  • Lemma 2.7
  • Definition 3.2
  • Lemma 3.3
  • proof
  • ...and 35 more