Implications

We have the following indirect implication of form equivalence classes:

30 \(\Rightarrow\) 336-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
30 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 378 clear
378 \(\Rightarrow\) 336-n clear

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

Howard-Rubin Number Statement
30:

Ordering Principle: Every set can be linearly ordered.

62:

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

378:

Restricted Choice for Families of Well Ordered Sets:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) such that the family of non-empty well orderable subsets of \(Y\) has a choice function.

336-n:

(For \(n\in\omega\), \(n\ge 2\).)  For every infinite set \(X\), there is an infinite \(Y \subseteq X\) such that the set of all \(n\)-element subsets of \(Y\) has a choice function.

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