Axiom Of Choice

We have the following indirect implication of form equivalence classes:

91 \(\Rightarrow\) 13 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
91 \(\Rightarrow\) 79	clear
79 \(\Rightarrow\) 94	clear
94 \(\Rightarrow\) 13	The Axiom of Choice, Jech, 1973b, page 148 problem 10.1

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

Howard-Rubin Number	Statement
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.
94:	\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals has a choice function. Jech [1973b], p 148 prob 10.1.
13:	Every Dedekind finite subset of \({\Bbb R}\) is finite.

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