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Comment to "Almost disjoint sets, the dense set problem and the partition calculus"

Júnio Luan Pereira

Abstract

This text highlights issues present in the proof of Lemma 6.10 of the Baumgartner (1943 -- 2011) article "Almost disjoint sets, the dense set problem and the partition calculus" of 1976, and intends to present a correction at the same time it proves a stronger result mentioned in the article to have similar proof.

Comment to "Almost disjoint sets, the dense set problem and the partition calculus"

Abstract

This text highlights issues present in the proof of Lemma 6.10 of the Baumgartner (1943 -- 2011) article "Almost disjoint sets, the dense set problem and the partition calculus" of 1976, and intends to present a correction at the same time it proves a stronger result mentioned in the article to have similar proof.
Paper Structure (3 sections, 1 theorem, 13 equations)

This paper contains 3 sections, 1 theorem, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Corrected version of Baumgartner's proof

Key Result

Proposition 2

Let $\mathfrak{M}$ be a countable transitive model of ZFC + GCH, and let $\kappa$ and $\lambda$ be cardinals in $\mathfrak{M}$ such that $\kappa$ is regular and $\kappa \leq \lambda$. If $\kappa$ is not inaccessible in $\mathfrak{M}$, then assume also that $\diamond_\kappa$ holds in $\mathfrak{M}$.

Theorems & Definitions (4)

  • Definition 1: baumgartner_almost-disjoint_1976
  • Proposition 2
  • proof : Proof for $\beta = \kappa$
  • proof : Proof for $\beta > \kappa$