# On the order dimension of locally countable partial orderings

## TL;DR

This paper determines the order dimension of certain infinite partial orders, showing it depends on the cardinality and cofinality, and explores implications for the Turing degrees' structure.

## Contribution

It establishes exact order dimension values for the partial order of finite subsets and bounds for locally countable partial orders, revealing new structural insights.

## Key findings

- Order dimension of finite subsets of 6 is 4(4(6))
- Order dimension of locally countable partial orders of size 4^+ is at most 4
- Turing degrees' dimension can be less than the continuum under ZFC

## 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.