Statement:
\((\forall n\in\omega - \{0\}) MC(\infty,\infty \), relatively prime to \(n\)): \(\forall n\in\omega -\{0\}\), if \(X\) is a set of non-empty sets, then there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a non-empty, finite subset of \(x\) and \(|f(x)|\) is relatively prime to \(n\).
Howard_Rubin_Number: 218
Parameter(s): This form does not depend on parameters
This form's transferability is: Not Transferable
This form's negation transferability is: Negation Transferable
Article Citations:
Bleicher-1965: Multiple choice axioms and the axiom of choice for finite sets
Book references
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 218 A | Existence of Complementary Subspaces: For every field \(F\), every vector space \(V\) over \(F\) has the property that if \(S\subseteq V\) is a subspace of \(V\), then there is a subspace \(S'\subseteq V\) such that \(S\cap S'= \{0\}\) and \(S\cup S'\) generates \(V\). H. Rubin/J. Rubin [1985], pp 123ff, Theorems 6.36, 6.37, and 6.38. |
Bleicher [1964]
Bleicher [1965]
Note [161] Book: Equivalents of the Axiom of Choice II |
| 218 B | For all primes \(p\), \(MC(\infty,\infty,\) relatively prime to \(p), \forall p, MC4(p)\): For every set \(X\) of non-empty sets there is a function \(f\) with domain \(X\) such that \(\forall u \in X\), \(f(u)\) is a finite non-empty subset of \(u\) such that \(|f(u)|\) and \(p\) are relatively prime. See Form 218, Rubin, H/Rubin, J. [1985], p.124. |
Bleicher [1964]
Note [161] Book: Equivalents of the Axiom of Choice II |