Implications

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We have the following indirect implication of form equivalence classes:

130 \(\Rightarrow\) 252 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
130 \(\Rightarrow\) 79	clear
79 \(\Rightarrow\) 252	consequence of the axiom of choice, Ash, C. J. 1975, J. Austral. Math. Soc. Ser. A.

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.
252:	The additive groups of \(\Bbb Q\oplus\Bbb R\) and \(\Bbb R\) are isomorphic.

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