Linear ROD subsets of Borel partial orders are countably cofinal in 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.

arXiv comment5 pages

References in corpus (2)

• When a partial Borel order is linearizable
• Linearization of analytic order relations

Cited by in corpus (7)

• Countable OD sets of reals belong to the ground model
• In Cohen generic extension, every countable OD set of reals belongs to the ground model
• OD elements of countable OD sets in the Solovay model
• On countable cofinality and decomposition of definable thin orderings
• Linearization of partial quasi-orderings in the Solovay model revisited
• Bounding and decomposing thin analytic partial orderings
• On countable cofinality of definable chains in Borel partial orders