Statement:
\(DC(\aleph_{\alpha})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(|Y|<\aleph_{\alpha}\), there is an \(x\in X\) with \(Y\mathrel R x\) then there is a function \(f:\aleph_{\alpha}\to X\) such that (\(\forall\beta < \aleph_{\alpha}\)) \(\{f(\gamma): \gamma < \beta\}\mathrel R f(\beta)\).
Howard_Rubin_Number: 87-alpha
Parameter(s): This form depends on the following parameter(s): \(\beta\), \(\beta\): ordinal number
This form's transferability is: Transferable
This form's negation transferability is: Negation Transferable
Article Citations:
Levy-1964: The interdependence of certain consequences of the axiom of choice
Book references
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 87A-alpha | \(Z_{\aleph_{\alpha }}\): Let \(P\) be a partially ordered set in which every well ordered chain has type \(< \aleph_{\alpha}\). If every well ordered chain in \(P\) has an upper bound in \(P\), then \(P\) contains a maximal element. |
Wolk [1983]
|
| 87B-alpha | \(Z^{*}_{\aleph_{\alpha}}\): Let \(P\) be a well-founded partially ordered set (that is, every chain in \(P\) is well ordered.) in which every chain has type \(< \aleph_{\alpha}\). If every chain in \(P\) has an upper bound in \(P\), then \(P\) contains a maximal element. |
Wolk [1983]
|