Axiom Of Choice

Statement:

\(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(|X| = \aleph_{\alpha }\), then \(X\) has a choice function.

Howard_Rubin_Number: 86-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:

The following forms are listed as conclusions of this form class in rfb1: 8, 43, 152, 196-alpha, 321, 344, 388, 106, 71-alpha, 86-alpha,

Howard-Rubin Number	Statement	References
86A-alpha	If \((P,\le)\) is a tree of height \(\lambda\le \aleph_\alpha\) which has fewer than \(\lambda\) branches of height \(<\lambda\) then one of the following holds for the upside down tree \((P,\ge)\): There is a cardinal \(\mu< \lambda\) such that \((P,\ge)\) does not contain a strong antichain of cardinality \(\mu\). \((P,\ge)\) contains a strong antichain of cardinality \(\lambda\). \((P,\ge)\) contains a chain of cardinality \(\lambda\).	Shannon [1992] Note [119]
86B-alpha	The frame envelope of any \(\aleph_\alpha\) frame is \(\aleph_\alpha\)-Lindelöf.	Banaschewski [1988] Note [29]