The Borel monadic theory of order is decidable

TL;DR

This paper proves that the monadic theory of the real line with order, restricted to Borel sets, is decidable, and extends results to larger classes of sets under certain hypotheses.

Contribution

It establishes the decidability of the Borel monadic theory of order and extends the result to larger classes of sets under determinacy assumptions.

Findings

The monadic theory of $(\mathbb R,\le)$ with Borel set quantification is decidable.

Boolean combinations of $F_\sigma$ sets form an elementary substructure.

Decidability extends to larger classes of sets under determinacy hypotheses.

Abstract

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. Under determinacy hypotheses, the proof extends to larger classes of sets.