Axiom Of Choice

We have the following indirect implication of form equivalence classes:

15 \(\Rightarrow\) 216 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
15 \(\Rightarrow\) 30	The Axiom of Choice, Jech, 1973b, page 53 problem 4.12
30 \(\Rightarrow\) 10	clear
10 \(\Rightarrow\) 216

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

Howard-Rubin Number	Statement
15:	\(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(\|A\|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 81(\(n\)).
30:	Ordering Principle: Every set can be linearly ordered.
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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