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The dense linear ordering principle

Published online by Cambridge University Press:  12 March 2014

David Pincus
David Pincus*
Affiliation:
Departments of Anesthesia, Harvard Medical School, Massachusetts General Hospital, Cambridge Hospital, 1493 Cambridge Street, Cambridge, MA 02139, USA, E-mail: pincus@helix.mgh.harvard.edu
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Abstract

Let DO denote the principle: Every infinite set has a dense linear ordering. DO is compared to other ordering principles such as O, the Linear Ordering principle, KW, the Kinna-Wagner Principle, and PI, the Prime Ideal Theorem, in ZF, Zermelo-Fraenkel set theory without AC, the Axiom of Choice.

The main result is:

Theorem. AC ⇒ KW ⇒ DO ⇒ O, and none of the implications is reversible in ZF + PI.

The first and third implications and their irreversibilities were known. The middle one is new. Along the way other results of interest are established. O, while not quite implying DO, does imply that every set differs finitely from a densely ordered set. The independence result for ZF is reduced to one for Fraenkel-Mostowski models by showing that DO falls into two of the known classes of statements automatically transferable from Fraenkel-Mostowski to ZF models. Finally, the proof of PI in the Fraenkel-Mostowski model leads naturally to versions of the Ramsey and Ehrenfeucht-Mostowski theorems involving sets that are both ordered and colored.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 2 , June 1997 , pp. 438 - 456
Copyright
Copyright © Association for Symbolic Logic 1997

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