Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
1 \(\Rightarrow\) 123

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

Howard-Rubin Number	Statement
1:	\(C(\infty,\infty)\): The Axiom of Choice: Every set of non-empty sets has a choice function.
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 \(x\).

Comment:

Back