On countable cofinality and decomposition of definable thin orderings

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 ${\Sigma }_2^1$ 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.

arXiv admin note: substantial text overlap with arXiv:1408.1202

References in corpus (4)

• 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

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