Statement:
\(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 81(\(n\)).
Howard_Rubin_Number: 15
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:
Kinna-Wagner-1955: Uber eine Abschwachung des Auswahlpostulates
Book references
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 15 A | \(K(1)\): \((\forall M)(\exists\alpha\) anordinal) (\(|M| = |A|\) for some \(A \subseteq {\cal P}(\alpha))\). (See Form 81\((n)\).) Jech [1973b] p 53 prob 4.12. |
Kinna-Wagner-1955
Book: The Axiom of Choice |
| 15 B | SDO: For any family \(X\) of infinite sets, there is afunction \(f\) with domain \(X\) such that \(\forall y\in X\), \(f(y)\) is a denselinear ordering of \(y\). Pincus [1997]. |
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| 15 C | SUO: For any family \(X\) of infinite sets, there is afunction \(f\) with domain \(X\) such that \(\forall y\in X\), \(f(y)\) is anunbounded linear ordering of \(y\). (An ordering is unbounded if thereis neither a least element nor a greatest element.) Pincus [1997]. |
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| 15 D | SDUO: For any family \(X\) of infinite sets, there is afunction \(f\) with domain \(X\) such that \(\forall y\in X\), \(f(y)\) isa dense unbounded linear ordering of \(y\). Pincus [1997].\medskip |
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