Axiom Of Choice

Statement:

\(E(I,III)\) (Howard/Yorke [1989]): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.)

Howard_Rubin_Number: 82

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:
Howard-Yorke-1989: Definitions of finite
Levy-1958: The independence of various definitions of finiteness

Book references

Note connections:
Note 94 Relationships between the different definitions of finite

The following forms are listed as conclusions of this form class in rfb1: 171, 253, 289, 13, 17, 18, 82, 83, 84, 98, 128, 144, 154, 198, 216, 358, 387, 65, 314, 79, 288-n, 342-n, 308-p, 373-n,

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Complete List of Equivalent Forms

Howard-Rubin Number	Statement	References
82 A	Every infinite set can be mapped onto \(\omega\).	Howard-Yorke-1989 Monro [1972]
82 B	\(E(Ia,III)\): For every set \(X\), if \(\cal P(X)\) is Dedekind finite then \(X\) is amorphous. \ac{Howard/Yorke} \cite{1989}, notes 57 and 94.	Howard-Yorke-1989 Note [57] Note [94]
82 C	\(P\)-\(\aleph_0\): For every infinite set \(X\), there is a partition of \(X\) of cardinality \(\aleph_0\). \ac{Gonz\'alez} \cite{1995a}.	Gonzalez [1995a]

Form Classes

Complete List of Equivalent Forms