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Combinatorial Properties Related to the Higher Baumgartner's Axiom

John Krueger

Abstract

We isolate two combinatorial properties, each expressible by a $Π_2$-sentence over the structure $(H(ω_3),\in,ω_1,ω_2,\text{NS}_{ω_2})$, such that each property is consistent with CH, and their conjunction together with $2^ω\le ω_2$ and $2^{ω_1} = 2^{ω_2} = ω_3$ implies the existence of a c.c.c. forcing which forces the higher Baumgartner's axiom.

Combinatorial Properties Related to the Higher Baumgartner's Axiom

Abstract

We isolate two combinatorial properties, each expressible by a -sentence over the structure , such that each property is consistent with CH, and their conjunction together with and implies the existence of a c.c.c. forcing which forces the higher Baumgartner's axiom.
Paper Structure (11 sections, 44 theorems, 17 equations)

This paper contains 11 sections, 44 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Combinatorial Properties
  3. The Single Forcing
  4. Adding the Bijections
  5. A Family of Models for Side Conditions
  6. The Definition of the Forcing Iteration
  7. Basic Facts About the Iteration
  8. Preserving Cardinals, 1
  9. Preparatory Lemmas
  10. Preserving Cardinals, 2
  11. Acknowledgements

Key Result

Theorem 2.2

Assume that $2^\omega \le \omega_2$, $2^{\omega_2} = \omega_3$, and $(**)$ holds. Then there exists a c.c.c. forcing which forces $\textsf{MA}_{\omega_1}$, $\textsf{BA}_{\omega_1}$, and $\textsf{BA}_{\omega_2}$.

Theorems & Definitions (91)

  • Definition 2.1: Moore and Todorčević (MT)
  • Theorem 2.2: Moore and Todorčević (MT)
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 81 more