Axiom Of Choice

We have the following indirect implication of form equivalence classes:

123 \(\Rightarrow\) 93 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
123 \(\Rightarrow\) 62	Two model theoretic ideas in independence proofs, Pincus, D. 1976, Fund. Math.
62 \(\Rightarrow\) 61	clear
61 \(\Rightarrow\) 88	clear
88 \(\Rightarrow\) 93	The Axiom of Choice, Jech, 1973b, page 7 problem 10

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

Howard-Rubin Number	Statement
123:	\(SPI^\): Uniform weak ultrafilter principle:* For each family \(F\) of infinite sets \(\exists f\) such that \(\forall x\in F\), \(f(x)\) is a non-principal ultrafilter on \(x\).
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.
88:	\(C(\infty ,2)\): Every family of pairs has a choice function.
93:	There is a non-measurable subset of \({\Bbb R}\).

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