Axiom Of Choice

We have the following indirect implication of form equivalence classes:

284 \(\Rightarrow\) 127 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
284 \(\Rightarrow\) 61	note-36
61 \(\Rightarrow\) 45-n	clear
45-n \(\Rightarrow\) 64	Classes of Dedekind finite cardinals, Truss, J. K. 1974a, Fund. Math.
64 \(\Rightarrow\) 127	Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung

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

Howard-Rubin Number	Statement
284:	A system of linear equations over a field \(F\) has a solution in \(F\) if and only if every finite sub-system has a solution in \(F\).
61:	\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element sets has a choice function.
45-n:	If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function.
64:	\(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)
127:	An amorphous power of a compact \(T_2\) space, which as a set is well orderable, is well orderable.

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