Implications

We have the following indirect implication of form equivalence classes:

123 \(\Rightarrow\) 33-n
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 \(\Rightarrow\) 61 clear
61 \(\Rightarrow\) 45-n clear
45-n \(\Rightarrow\) 33-n clear

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.

61:

\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element  sets has a choice function.

45-n:

If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function.

33-n:

If \(n\in\omega-\{0,1\}\), \(C(LO,n)\):  Every linearly ordered set of \(n\) element sets has  a choice function.

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