You are not currently logged in.
Access JSTOR through your library or other institution:
If You Use a Screen ReaderThis content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
The Dense Linear Ordering Principle
The Journal of Symbolic Logic
Vol. 62, No. 2 (Jun., 1997), pp. 438-456
Published by: Association for Symbolic Logic
Stable URL: http://www.jstor.org/stable/2275540
Page Count: 19
You can always find the topics here!Topics: Infinite sets, Lexicography, Logical theorems, Boolean algebras, Isomorphism, Axiom of choice, Mathematical theorems, Mathematical set theory, Automorphisms, Symbolism
Were these topics helpful?See something inaccurate? Let us know!
Select the topics that are inaccurate.
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Preview not available
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.
$AC \Longrightarrow KW \Longrightarrow DO \Longrightarrow O$, 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.
The Journal of Symbolic Logic © 1997 Association for Symbolic Logic