On countable cofinality and decomposition of definable thin orderings
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 Σ¹₂ thin sets under the assumption that for all reals x. We also prove that definable thin wellorderings admit partitions into definable chains in the Solovay model.