Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
107 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 61 clear
61 \(\Rightarrow\) 11 clear
11 \(\Rightarrow\) 12 clear
12 \(\Rightarrow\) 336-n clear
336-n \(\Rightarrow\) 64 Weak choice principles, De la Cruz, O. 1998a, Proc. Amer. Math. Soc.

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.

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.

11:

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\), \(A\) has an infinite subset \(B\) such that for every \(n\in\omega\), \(n>0\), the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco [1998b]

12:

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\) and every \(n\in\omega\), there is an infinite subset \(B\) of \(A\) such the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco} [1998b]

336-n:

(For \(n\in\omega\), \(n\ge 2\).)  For every infinite set \(X\), there is an infinite \(Y \subseteq X\) such that the set of all \(n\)-element subsets of \(Y\) 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.)

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