Statement:
\(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function.
Howard_Rubin_Number: 62
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:
Book references
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 62 A | (Depends on the two primes \(p_1\ne p_2\)) Form 308 \((p_1\)) + Form 308 (\(p_2\)). |
Howard-Yorke-1987
|
| 62 B | (Depends on the prime \(p_1\)) Form 45 (\(p_1\)) + Form 308 (\(p_1\)). |
Howard-Yorke-1987
|
| 62 C | For every prime \(p\), if \(\{\,G_y : y\in Y\,\}\) is a set of finite groups, then the weak direct product \(\prod_{y\in Y} G_y\) has a maximal \(p\)-subgroup. \ac{Howard/Yorke} \cite{1987} and note 24. |
Howard-Yorke-1987
Note [24] |
| 62 D | For every prime \(p\), if \(Y\) is a set of non-empty, finite sets, then the weak direct product \(\prod_{y\in Y} S_y\) has a maximal \(p\)-subgroup. (\(S_y\) is the symmetric group on \(y\).) |
Howard-Yorke-1987
Note [24] |
| 62 E | \(KW(\infty,<\aleph_0)\), The Kinna-Wagner Selection Principle for families of finite sets: For every set \(M\) of finite sets there is a function \(f\) such that for all \(A \in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). |
Note [70] |