Implications

We have the following indirect implication of form equivalence classes:

130 \(\Rightarrow\) 108
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
130 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 38 clear
38 \(\Rightarrow\) 108 clear

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

Howard-Rubin Number Statement
130:

\({\cal P}(\Bbb R)\) is well orderable.

79:

\({\Bbb R}\) can be well ordered.  Hilbert [1900], p 263.

38:

\({\Bbb R}\) is not the union of a countable family of countable sets.

108:

There is an ordinal \(\alpha\) such that \(2^{\aleph _{\alpha}}\) is not the union of a denumerable set of denumerable sets.

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