Axiom Of Choice

We have the following indirect implication of form equivalence classes:

346 \(\Rightarrow\) 64 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 \(\Rightarrow\) 83	Definitions of finite, Howard, P. 1989, Fund. Math.
83 \(\Rightarrow\) 64	The Axiom of Choice, Jech, 1973b, page 52 problem 4.10

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.)
83:	\(E(I,II)\) Howard/Yorke [1989]: \(T\)-finite is equivalent to finite.
64:	\(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)

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