Axiom Of Choice

We have the following indirect implication of form equivalence classes:

15 \(\Rightarrow\) 48-K 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 \(\Rightarrow\) 62	clear
62 \(\Rightarrow\) 121	clear
121 \(\Rightarrow\) 122	clear
122 \(\Rightarrow\) 48-K	clear

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.
62:	\(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function.
121:	\(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function.
122:	\(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function.
48-K:	If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(WO,K)\): For every \(n\in K,\) \(C(WO,n)\).

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