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The Borel monadic theory of order is decidable

Sven Manthe

Abstract

The monadic theory of $(\mathbb R,\le)$ with quantification restricted to Borel sets is decidable. The Boolean combinations of $F_σ$ sets form an elementary substructure of the Borel sets. Under determinacy hypotheses, the proof extends to larger classes of sets.

The Borel monadic theory of order is decidable

Abstract

The monadic theory of with quantification restricted to Borel sets is decidable. The Boolean combinations of sets form an elementary substructure of the Borel sets. Under determinacy hypotheses, the proof extends to larger classes of sets.
Paper Structure (35 sections, 68 theorems, 28 equations)

This paper contains 35 sections, 68 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Proof outline
  3. Consequences
  4. Definability in the Borel monadic theory of order
  5. The Baire property
  6. Notation and prerequisites
  7. Descriptive set theory
  8. Order theory
  9. Ramsey theory
  10. Logic
  11. Constructing Cantor sets
  12. Cantor subsets
  13. Iso-uniform orders
  14. Definability in restricted monadic theories
  15. Shelah's decidability technique
  16. ...and 20 more sections

Key Result

Theorem 1.1

The monadic theory of $(\mathbb R,\le)$ with quantification restricted to Borel sets is decidable. The Boolean combinations of $F_\sigma$ sets form an elementary substructure of the Borel sets.

Theorems & Definitions (171)

  • Theorem 1.1
  • Theorem 1.2: ZF+Dependent choices
  • Definition 1.3
  • Corollary 1.4: Projective determinacy
  • Corollary 1.5: ZF+Dependent choices+Determinacy
  • Proposition 1.6
  • proof
  • Corollary 1.7
  • proof
  • Conjecture 1.8
  • ...and 161 more