Linear ROD subsets of Borel partial orders are countably cofinal in Solovay's model

Details Linear ROD subsets of Borel partial orders are countably cofinal in Solovay's model Linear ROD subsets of Borel partial orders are countably cofinal in Solovay's model

2010-05-30

arxiv.org/abs/1005.5534v2

Mathematics

Logic

5 pages

Scientific paper

The following is true in the Solovay model. 1. If $\leq$ is a Borel partial quasi-order on a Borel set $D$ of the reals, $X$ is a ROD subset of $D$, and $\leq$ restricted to $X$ is linear, then $X$ is countably cofinal in the sense of $\leq$. 2. If in addition every countable set $Y$ of $D$ has a strict upper bound in the sense of $\leq$, then the ordering $< D ; \leq >$ has no maximal chains that are ROD sets.

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