Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
328 \(\Rightarrow\) 126	clear
126 \(\Rightarrow\) 94	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
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
328:	\(MC(WO,\infty)\): For every well ordered set \(X\) such that for all \(x\in X\), \(\|x\|\ge 1\), there is a function \(f\) such that and for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). (See Form 67.)
126:	\(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
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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