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arXiv · 1902.06030

On the order dimension of locally countable partial orderings

Kojiro Higuchi·Steffen Lempp·Diip Raghavan·Frank Stephan
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Abstract

We show that the order dimension of the partial order of all finite subsets of $\kappa$ under set inclusion is ${\log}_{2}({\log}_{2}(\kappa))$ whenever $\kappa$ is an infinite cardinal. We also show that the order dimension of any locally countable partial ordering $(P, <)$ of size $\kappa^+$, for any $\kappa$ of uncountable cofinality, is at most $\kappa$. In particular, this implies that it is consistent with ZFC that the dimension of the Turing degrees under partial ordering can be strictly less than the continuum.

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Kojiro Higuchi, Steffen Lempp, Diip Raghavan, Frank Stephan. 2019-02-16. On the order dimension of locally countable partial orderings. https://arxiv.org/abs/1902.06030

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