On countable cofinality and decomposition of definable thin orderings

TL;DR

This paper investigates the properties of definable thin sets within Borel partial orderings, establishing conditions under which they are countably cofinal and can be partitioned into definable chains, with results applicable in various models.

Contribution

It proves that certain definable thin sets are necessarily countably cofinal and that definable thin wellorderings can be partitioned into definable chains, extending understanding of their structure.

Findings

Definable thin sets in some models are countably cofinal.

Definable thin wellorderings can be partitioned into definable chains.

Results apply to analytic, ROD in Solovay, and certain $oldsymbol{ m oldsymbol{ m extit{ extSigma}}^1_2}$ sets.

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 $\Sigma^1_2$ thin sets in the assumption that $\omega_1^{L[x]}<\omega_1$ for all reals $x$. We also prove that definable thin wellorderings admit partitions into definable chains in the Solovay model.