Axiom Of Choice

We have the following indirect implication of form equivalence classes:

346 \(\Rightarrow\) 84 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
346 \(\Rightarrow\) 126	The vector space Kinna-Wagner Principle is equivalent to the axiom of choice, Keremedis, K. 2001a, Math. Logic Quart.
126 \(\Rightarrow\) 82	note-76
82 \(\Rightarrow\) 84	Definitions of finite, Howard, P. 1989, Fund. Math.

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

Howard-Rubin Number	Statement
346:	If \(V\) is a vector space without a finite basis then \(V\) contains an infinite, well ordered, linearly independent subset.
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 \(y\).
82:	\(E(I,III)\) (Howard/Yorke [1989]): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.)
84:	\(E(II,III)\) (Howard/Yorke [1989]): \((\forall x)(x\) is \(T\)-finite if and only if \(\cal P(x)\) is Dedekind finite).

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