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Iterated club shooting and the stationary-logic constructible model

Ur Ya'ar

Abstract

We investigate iterating the construction of $C(\mathtt{aa})$, the $L$-like inner model constructed using stationary-logic. We show that it is possible to force over generic extensions of $L$ to obtain a model of $V=C(\mathtt{aa})$, and to obtain models in which the sequence of iterated $C(\mathtt{aa})$s is decreasing of arbitrarily large order types. For this we prove distributivity and stationary-set preservation properties for countable iterations of club-shooting forcings using mutually stationary sets, and introduce the notion of mutually fat sets which yields better distributivity results even for uncountable iterations.

Iterated club shooting and the stationary-logic constructible model

Abstract

We investigate iterating the construction of , the -like inner model constructed using stationary-logic. We show that it is possible to force over generic extensions of to obtain a model of , and to obtain models in which the sequence of iterated s is decreasing of arbitrarily large order types. For this we prove distributivity and stationary-set preservation properties for countable iterations of club-shooting forcings using mutually stationary sets, and introduce the notion of mutually fat sets which yields better distributivity results even for uncountable iterations.
Paper Structure (15 sections, 25 theorems, 59 equations, 1 figure)

This paper contains 15 sections, 25 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction and preliminaries
  2. Introduction
  3. Stationary sets
  4. Stationary logic and its constructible model
  5. Intersections of forcing extensions
  6. Iterated club shooting
  7. Shooting one club
  8. Shooting countably many clubs
  9. Shooting uncountably many clubs
  10. $\square$-sequence on singulars
  11. Forcing non-reflecting stationary sets
  12. Coding by shooting clubs
  13. Coding a set into a model of $V=C(\mathop{\mathrm{\mathtt{aa}}}\nolimits)$
  14. Iterating $C(\mathop{\mathrm{\mathtt{aa}}}\nolimits)$
  15. Open questions

Key Result

Lemma 1.2

Let $\kappa>\omega$ be regular, $A\subseteq E_{\omega}^{\kappa}$, $B\subseteq\mathcal{P}_{\omega_{1}}(\kappa)$.

Figures (1)

  • Figure 1: A descending $C(\mathop{\mathrm{\mathtt{aa}}}\nolimits)$ sequence of length $\omega$

Theorems & Definitions (73)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Lemma 1.6
  • proof
  • ...and 63 more