Implications

We have the following indirect implication of form equivalence classes:

344 \(\Rightarrow\) 80
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
344 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 80 clear

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

Howard-Rubin Number Statement
344:

If \((E_i)_{i\in I}\) is a family of non-empty sets, then there is a family \((U_i)_{i\in I}\) such that \(\forall i\in I\), \(U_i\) is an ultrafilter on \(E_i\).

62:

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

10:

\(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function.

80:

\(C(\aleph_{0},2)\):  Every denumerable set of  pairs has  a  choice function.

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