Axiom Of Choice

We have the following indirect implication of form equivalence classes:

28-p \(\Rightarrow\) 5 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
28-p \(\Rightarrow\) 427	clear
427 \(\Rightarrow\) 67	clear
67 \(\Rightarrow\) 89	On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
89 \(\Rightarrow\) 90	The Axiom of Choice, Jech, 1973b, page 133
90 \(\Rightarrow\) 51	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
51 \(\Rightarrow\) 337	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
337 \(\Rightarrow\) 92	clear
92 \(\Rightarrow\) 94	clear
94 \(\Rightarrow\) 5	clear

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

Howard-Rubin Number	Statement
28-p:	(Where \(p\) is a prime) AL20(\(\mathbb Z_p\)): Every vector space \(V\) over \(\mathbb Z_p\) has the property that every linearly independent subset can be extended to a basis. (\(\mathbb Z_p\) is the \(p\) element field.) Rubin, H./Rubin, J. [1985], p. 119, Statement AL20
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).
89:	Antichain Principle: Every partially ordered set has a maximal antichain. Jech [1973b], p 133.
90:	\(LW\): Every linearly ordered set can be well ordered. Jech [1973b], p 133.
51:	Cofinality Principle: Every linear ordering has a cofinal sub well ordering. Sierpi\'nski [1918], p 117.
337:	\(C(WO\), uniformly linearly ordered): If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\).
92:	\(C(WO,{\Bbb R})\): Every well ordered family of non-empty subsets of \({\Bbb R}\) has a choice function.
94:	\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals has a choice function. Jech [1973b], p 148 prob 10.1.
5:	\(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function.

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Implications