Axiom Of Choice

Statement:

Cofinality Principle: Every linear ordering has a cofinal sub well ordering. Sierpi\'nski [1918], p 117.

Howard_Rubin_Number: 51

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:
Sierpi'nski-1918: L’axiome de M. Zermelo et son rˆole dans la th´eorie des ensembles et l’analyse

Book references

Note connections:

The following forms are listed as conclusions of this form class in rfb1: 99, 77, 25, 40, 49, 51, 91, 144, 208, 294, 316, 337,

Howard-Rubin Number	Statement	References
51 A	For all linear orders \((X,\le )\), there is a \(\sup\) function on the bounded, well ordered subsets of \(X\).	Manka [1988a] Note [38]
51 B	\(Z(L,W)\): Every non-empty linearly ordered set in which every well ordered subset has an upper bound has a maximal element. H. Rubin/J. Rubin [1985], page 37, \(M1(L,W)\).	Harper-Rubin-1976 Manka [1988a] Note [39] Book: Equivalents of the Axiom of Choice II
51 C	For every linearly ordered set \((X,\le )\), if every well ordered subset of \(X\) is bounded above then there is a \(\sup\) function on the well ordered subsets of \(X\).	Manka [1988a] Note [38]