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Equivalence of generics

Iian B. Smythe

Abstract

Given a countable transitive model of set theory and a partial order contained in it, there is a natural countable Borel equivalence relation on generic filters over the model; two are equivalent if they yield the same generic extension. We examine the complexity of this equivalence relation for various partial orders, with particular focus on Cohen and random forcing. We prove, amongst other results, that the former is an increasing union of countably many hyperfinite Borel equivalence relations, while the latter is neither amenable nor treeable.

Equivalence of generics

Abstract

Given a countable transitive model of set theory and a partial order contained in it, there is a natural countable Borel equivalence relation on generic filters over the model; two are equivalent if they yield the same generic extension. We examine the complexity of this equivalence relation for various partial orders, with particular focus on Cohen and random forcing. We prove, amongst other results, that the former is an increasing union of countably many hyperfinite Borel equivalence relations, while the latter is neither amenable nor treeable.

Paper Structure

This paper contains 7 sections, 31 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. General results
  3. Spaces of generics
  4. Automorphisms and homogeneity
  5. Cohen reals
  6. Random reals
  7. Further questions

Key Result

Lemma 2.2

$\mathrm{Gen}^\mathbb{P}_M$ is a $G_\delta$ subset of $2^\mathbb{P}$, and thus a Polish space.

Theorems & Definitions (57)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 47 more