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Combinatorial Properties of the Raisonnier Filter

Spyridon Dialiatsis, Yurii Khomskii

Abstract

The Raisonnier Filter is a combinatorial object isolated by Jean Raisonnier in order to simplify Shelah's proof that if all $\boldsymbolΣ^1_3$ sets are Lebesgue-measurable then there is an inner model with an inaccessible cardinal. In this paper, we study the combinatorics of a general version of the Raisonnier filter, with an eye to potential applications in descriptive set theory. Among the most interesting of our results is a partial converse to Raisonnier's theorem, which can be used to provide a new characterisation of the statement "all $\boldsymbolΣ^1_2$ sets are measurable". We also introduce an ideal on the Cantor Space induced by the Raisonnier filter and study its cardinal characteristics, connecting them to the well-known characteristics in Cichoń's Diagram.

Combinatorial Properties of the Raisonnier Filter

Abstract

The Raisonnier Filter is a combinatorial object isolated by Jean Raisonnier in order to simplify Shelah's proof that if all sets are Lebesgue-measurable then there is an inner model with an inaccessible cardinal. In this paper, we study the combinatorics of a general version of the Raisonnier filter, with an eye to potential applications in descriptive set theory. Among the most interesting of our results is a partial converse to Raisonnier's theorem, which can be used to provide a new characterisation of the statement "all sets are measurable". We also introduce an ideal on the Cantor Space induced by the Raisonnier filter and study its cardinal characteristics, connecting them to the well-known characteristics in Cichoń's Diagram.
Paper Structure (7 sections, 24 theorems, 25 equations, 2 figures)

This paper contains 7 sections, 24 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions and Basic Context
  3. Simple properties of the Raisonnier filter
  4. A characterization of the rapidity of $\mathcal{F}_X$
  5. A partial converse to Raisonnier's theorem
  6. The Raisonnier Ideal and its cardinal characteristics
  7. Conclusions and Open Questions

Key Result

Lemma 2.3

Let $\mathcal{F}$ be a filter and let $\varphi: \omega \to \omega$ be an increasing, unbounded function such that $\varphi(0) = 0$. Then the following are equivalent:

Figures (2)

  • Figure 1: A real $x \in 2^\omega$ translated into $\mathop{\mathrm{nat}}\nolimits_d(x) \in \omega^\omega$, using the partitioning real $d = (0,4,6,9,14,15,18,\dots)$.
  • Figure 2: A real $f \in \omega^\omega$ translated into $\mathop{\mathrm{bin}}\nolimits_d(x) \in 2^\omega$, using the partitioning real $d = (0,4,6,9,14,15,18,\dots)$.

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Theorem 2.5: Solovay/Bartoszyński
  • proof
  • Definition 2.6
  • Definition 2.7: The Raisonnier Filter
  • Theorem 2.8: Raisonnier, RaisonnierFilter
  • ...and 49 more