Axiom Of Choice

We have the following indirect implication of form equivalence classes:

107 \(\Rightarrow\) 132 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
107 \(\Rightarrow\) 62	clear
62 \(\Rightarrow\) 132	Sequential compactness and the axiom of choice, Brunner, N. 1983b, Notre Dame J. Formal Logic

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

Howard-Rubin Number	Statement
107:	M. Hall's Theorem: Let \(\{S(\alpha): \alpha\in A\}\) be a collection of finite subsets (of a set \(X\)) then if () for each finite \(F \subseteq A\) there is an injective choice function on \(F\) then there is an injective choice function on \(A\). (That is, a 1-1 function \(f\) such that \((\forall\alpha\in A)(f(\alpha)\in S(\alpha))\).) (According to a theorem of P. Hall (\(\)) is equivalent to \(\left \|\bigcup_{\alpha\in F} S(\alpha)\right\|\ge \|F\|\). P. Hall's theorem does not require the axiom of choice.)
62:	\(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function.
132:	\(PC(\infty, <\aleph_0,\infty)\): Every infinite family of finite sets has an infinite subfamily with a choice function.

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