1 results
Bi-orders do not arise from total orders
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 65 / Issue 1 / March 2022
- Published online by Cambridge University Press:
- 20 April 2021, pp. 217-224
- Print publication:
- March 2022
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- Article
- Export citation
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We present a Zermelo–Fraenkel (
$\textbf {ZF}$
) consistency result regarding bi-orderability of groups. A classical consequence of the ultrafilter lemma is that a group is bi-orderable if and only if it is locally bi-orderable. We show that there exists a model of 
$\textbf {ZF}$
plus dependent choice in which there is a group which is locally free (ergo locally bi-orderable) and not bi-orderable, and the group can be given a total order. The model also includes a torsion-free abelian group which is not bi-orderable but can be given a total order.
