Implications

We have the following indirect implication of form equivalence classes:

30 \(\Rightarrow\) 342-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
30 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 132 Sequential compactness and the axiom of choice, Brunner, N. 1983b, Notre Dame J. Formal Logic
132 \(\Rightarrow\) 342-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.

132:

\(PC(\infty, <\aleph_0,\infty)\):  Every infinite family of finite  sets has an infinite subfamily with a choice function.

342-n:

(For \(n\in\omega\), \(n\ge 2\).) \(PC(\infty,n,\infty)\):  Every infinite family of \(n\)-element sets has an infinite subfamily with a choice function. (See Form 166.)

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