Implications

We have the following indirect implication of form equivalence classes:

123 \(\Rightarrow\) 216
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 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 216

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.

10:

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

216:

Every infinite tree has either an infinite chain or an infinite antichain.

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