On countable cofinality and decomposition of definable thin orderings
Volume 235 / 2016
Fundamenta Mathematicae 235 (2016), 13-36
MSC: Primary 03E15; Secondary 03E35.
DOI: 10.4064/fm977-10-2015
Published online: 11 May 2016
Abstract
We prove that in some cases definable thin sets (including chains) of Borel partial orderings are necessarily countably cofinal. This includes the following cases: analytic thin sets, ROD thin sets in the Solovay model, and ${\mathbf \Sigma }_{2}^{1}$ thin sets under the assumption that $\omega _{1}^{{\bf L}[x]} \lt \omega _{1}$ for all reals $x$. We also prove that definable thin wellorderings admit partitions into definable chains in the Solovay model.
