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

TL;DR

In Solovay's model, Borel partial orders with ROD subsets that are linearly ordered are always countably cofinal, and under certain conditions, these orders lack ROD maximal chains, revealing structural properties of definable orders.

Contribution

This paper proves that in Solovay's model, ROD subsets of Borel partial orders that are linearly ordered are necessarily countably cofinal, a new insight into the structure of definable orders.

Findings

ROD subsets linearly ordered are countably cofinal in Solovay's model

No ROD maximal chains exist under certain conditions in Borel partial orders

Structural properties of definable orders in Solovay's model

Abstract

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