Axiom Of Choice

We have the following indirect implication of form equivalence classes:

123 \(\Rightarrow\) 132 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 \(\Rightarrow\) 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
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.
132:	\(PC(\infty, <\aleph_0,\infty)\): Every infinite family of finite sets has an infinite subfamily with a choice function.

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