Axiom Of Choice

We have the following indirect implication of form equivalence classes:

$\Rightarrow$ 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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