Linear ROD subsets of Borel partial orders are countably cofinal in the Solovay model

TL;DR

In the Solovay model, Borel partial orders restricted to ROD linearly ordered subsets are countably cofinal, and under certain conditions, these orders lack maximal ROD chains.

Contribution

The paper proves that in the Solovay model, ROD subsets of Borel partial orders have countably cofinal restrictions and identifies conditions for the absence of maximal ROD chains.

Findings

ROD subsets of Borel partial orders are countably cofinal in the Solovay model.

If every countable set has a strict upper bound, no maximal ROD chains exist.

Results are specific to the properties of the Solovay model.

Abstract

The following is true in the Solovay model. 1. If $\le$ is a Borel partial order on a set $D$ of the reals, and $X$ is a ROD subset of $D$ linearly ordered by $\le$, then the restriction of $\le$ onto $X$ is countably cofinal. 2. If in addition every countable set $Y$ of $D$ has a strict upper bound in the sense of $\le$ then the ordering $< D ; \le >$ has no maximal chains that are ROD sets.