Statement:
Every set is almost well orderable.
Howard_Rubin_Number: 144
Parameter(s): This form does not depend on parameters
This form's transferability is: Unknown
This form's negation transferability is: Negation Transferable
Article Citations:
Keisler-1970: Logic with the quantifier ``there exist uncountably many''
Book references
Note connections:
Note 25
For forms Form 144 and Form 179-\(\epsilon\)
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 144 A | Every \(p\) space is discrete. (A \(p\) space is a\(T_2\) space in which the intersection of any well orderable familyof open sets is open). Brunner [1984c]. |
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| 144 B | Set Induction Principle. If \(X\subseteq \cal P(S)\) contains all finite subsets of \(S\) and forevery subset \(Y\subseteq X\) such that \(Y\) is well orderedby \(\subseteq\), \(\bigcup Y\in X\)(such an \(X\) is called \(\cal W\)-inductive in Ern\'e [2000])then \(S\in X\). Ern\'e [2000], notes 154 and 156. |
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| 144 C | Every \(\cal W\)-inductive system is \(^c\calD\)-inductive. Ern\'e [2000] and Note 154. |
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| 144 D | Every \(\cal W\)-inductive decreasing system(i.e., closed under \(\subseteq\)) is of finite character.Ern\'e [2000] and Note 154. |
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| 144 E | Every \(\cal W\)-union preserving map on apower set preserves \(\cal D\)-unions. Ern\'e [2000] and note154. |
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| 144 F | Every \(\cal W\)-inductive closure system isan algebraic lattice (using the ordering \(\subseteq\)).Ern\'e [2000] and Note 154. |
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| 144 G | Every \(\cal W\)-complete \(\lor\)-semilatticewith least element is a complete lattice. Ern\'e [2000] andNote 154. |
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| 144 H | If \(X\) is a \(\cal W\)-subcomplete subset of acomplete lattice and \(X\) is closed under \(\le\), then \(X\) is Scottclosed. Ern\'e [2000] and Note 154. |
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| 144 I | Every \(\cal W\)-compact element of a completelattice is compact. Ern\'e [2000] and Note 154. |
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| 144 J | Every \(\cal W\)-surpema preserving map oncomplete lattices preserves \(\cal D\)-suprema.Ern\'e [2000] and Note 154. |
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| 144 K | Every \(\cal W\)-complete poset is \(^c\calD\)-complete. Ern\'e [2000] and Note 154. |
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| 144 L | Every \(\cal W\)-subcomplete subset of a posetis \(^c\cal D\)-subcomplete. Ern\'e [2000] and Note 154.Ern\'e [2000] and Note 154. |
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| 144 M | Every $\Cal W$-suprema preserving map on $\Cal W$-complete posets preserves $^c\Cal D$-surpema. \ac{Ern\'e} \cite{2000} and note 154. |
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