Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
90 \(\Rightarrow\) 91	The Axiom of Choice, Jech, 1973b, page 133
91 \(\Rightarrow\) 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
90:	\(LW\): Every linearly ordered set can be well ordered. Jech [1973b], p 133.
91:	\(PW\): The power set of a well ordered set can be well ordered.
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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