Axiom Of Choice

We have the following indirect implication of form equivalence classes:

95-F \(\Rightarrow\) 396 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
95-F \(\Rightarrow\) 67	Some theorems on vector spaces and the axiom of choice, Bleicher, M. 1964, Fund. Math. The Axiom of Choice, Jech, 1973b, page 148 problem 10.4
67 \(\Rightarrow\) 112	clear
112 \(\Rightarrow\) 395	clear
395 \(\Rightarrow\) 396	clear

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

Howard-Rubin Number	Statement
95-F:	Existence of Complementary Subspaces over a Field \(F\): If \(F\) is a field, then every vector space \(V\) over \(F\) has the property that if \(S\subseteq V\) is a subspace of \(V\), then there is a subspace \(S'\subseteq V\) such that \(S\cap S'= \{0\}\) and \(S\cup S'\) generates \(V\). H. Rubin/J. Rubin [1985], pp 119ff, and Jech [1973b], p 148 prob 10.4.
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).
112:	\(MC(\infty,LO)\): For every family \(X\) of non-empty sets each of which can be linearly ordered there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
395:	\(MC(LO,LO)\): For each linearly ordered family of non-empty linearly orderable sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is a non-empty, finite subset of \(x\).
396:	\(MC(LO,WO)\): For each linearly ordered family of non-empty well orderable sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is a non-empty, finite subset of \(x\).

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