Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
123 \(\Rightarrow\) 123	On a question of Tarski and a maximal theorem of Kurepa, Halpern, J.D. 1972, Pacific J. Math.

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 \(x\).
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 \(x\).

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