Axiom Of Choice

We have the following indirect implication of form equivalence classes:

335-n \(\Rightarrow\) 390 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
335-n \(\Rightarrow\) 333	Bases for vector spaces over the two element field and the axiom of choice, Keremedis, K. 1996a, Proc. Amer. Math. Soc.
333 \(\Rightarrow\) 67	clear
67 \(\Rightarrow\) 126	clear
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
64 \(\Rightarrow\) 390	clear

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

Howard-Rubin Number	Statement
335-n:	Every quotient group of an Abelian group each of whose elements has order \(\le n\) has a set of representatives.
333:	\(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of sets such that for all \(x\in X\), \(\|x\|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(\|f(x)\|\) is odd.
67:	\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).
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.)
390:	Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements. Ash [1983].

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