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Ideal Analytic sets

Łukasz Mazurkiewicz, Szymon Żeberski

Abstract

The aim of this paper is to give natural examples of $\mathbfΣ_1^1$-complete and $\mathbfΠ_1^1$-complete sets. In the first part, we consider ideals on $ω$. In particular, we show that the Hindman ideal $\mathcal{H}$ is $\mathbfΠ_1^1$-complete and consider a number of ideals generated in the similar fashion. Moreover, we show that the ideal $\mathcal{D}$ is also $\mathbfΠ_1^1$-complete. In the second part, we focus on families of trees (on $ω$ and $2$) containing a specific tree type. We show the connection between two topics and explore some classical tree types (like Sacks and Miller).

Ideal Analytic sets

Abstract

The aim of this paper is to give natural examples of -complete and -complete sets. In the first part, we consider ideals on . In particular, we show that the Hindman ideal is -complete and consider a number of ideals generated in the similar fashion. Moreover, we show that the ideal is also -complete. In the second part, we focus on families of trees (on and ) containing a specific tree type. We show the connection between two topics and explore some classical tree types (like Sacks and Miller).
Paper Structure (8 sections, 17 theorems, 45 equations)

This paper contains 8 sections, 17 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Ideals on $\omega$
  3. Unified approach to reduction
  4. Ramsey ideal
  5. Hindman ideal
  6. Modifications of the Hindman ideal
  7. Differences instead of sums
  8. Trees and ideals

Key Result

Theorem 2.1

$\mathcal{R}$ is $\mathop{\mathrm{\mathbf{\Pi}_1^1}}\nolimits$-complete.

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 2.1
  • proof
  • Definition 7
  • Theorem 2.2
  • ...and 31 more