Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
68 \(\Rightarrow\) 62	Subgroups of a free group and the axiom of choice, Howard, P. 1985, J. Symbolic Logic
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
68:	Nielsen-Schreier Theorem: Every subgroup of a free group is free. Jech [1973b], p 12.
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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