Statement:
Ordering Principle: Every set can be linearly ordered.
Howard_Rubin_Number: 30
Parameter(s): This form does not depend on parameters
This form's transferability is: Transferable
This form's negation transferability is: Negation Transferable
Article Citations:
Schroder-1898: Über zwei Definitionen der Endlichkeit und G. Cantor’sche Sätze,
Book references
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 30 A | If \(R\) is a partial ordering in which the set of predecessors of each element is linearly ordered, then there is a linear ordering extending \(R\). |
Felgner [1969]
|
| 30 B | ADO: Every infinite set differs finitely from a set with a dense linear ordering: for all infinite \(x\), there is a \(y\subseteq x\) such that \(y\) has a dense linear ordering and \(x-y\) is finite. |
Pincus [1997]
|
| 30 C | SADO: For every set \(F\) there is a function \(f\) with domain \(F\) such that for all \(x\in F\), \(f(x)\subseteq x\), \(f(x)\) has a dense linear ordering, and \(x - f(x)\) is finite. |
Pincus [1997]
|
| 30 D | UO: There exists an unbounded linear ordering on every infinite set. (An ordering is unbounded if there is neither a least element nor a greatest element.) |
Pincus [1997]
|
| 30 E | ADUO: For all infinite \(x\), there exists \(y\subseteq x\) such that \(y\) has a dense unbounded linear ordering and \(x - y\) is finite. |
Pincus [1997]
|
| 30 F | SADUO: For every set \(F\) there is a function \(f\) with domain \(F\) such that for all \(x\in F\), \(f(x)\subseteq x\), \(f(x)\) has a dense unbounded linear ordering, and \(x - f(x)\) is finite. |
Pincus [1997]
|
| 30 G | Weak Hausdorff Lemma: For every set \(X\), every \(\subseteq\)-chain in \(\cal P(X)\) is contained in a maximal \(\subseteq\)-chain. |
Note [130] |