Implications

We have the following indirect implication of form equivalence classes:

427 \(\Rightarrow\) 64
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
427 \(\Rightarrow\) 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

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

Howard-Rubin Number Statement
427: \(\exists F\) AL20(\(F\)): There is a field \(F\) such that every vector space \(V\) over \(F\) has the property that every independent subset of \(V\) can be extended to a basis.  \ac{Bleicher} \cite{1964}, \ac{Rubin, H.\/Rubin, J \cite{1985, p.119, AL20}.
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.)

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