Implications

We have the following indirect implication of form equivalence classes:

39 \(\Rightarrow\) 27
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
39 \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 27 clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
39:

\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.

8:

\(C(\aleph_{0},\infty)\):

27:

\((\forall \alpha)( UT(\aleph_{0},\aleph_{\alpha}, \aleph_{\alpha}))\): The  union of denumerably many sets each of power \(\aleph_{\alpha }\) has power \(\aleph_{\alpha}\). Moore, G. [1982], p 36.

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