Axiom Of Choice

We have the following indirect implication of form equivalence classes:

15 \(\Rightarrow\) 280 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 \(\Rightarrow\) 61	clear
61 \(\Rightarrow\) 88	clear
88 \(\Rightarrow\) 142	The Axiom of Choice, Jech, 1973b, page 7 problem 11
142 \(\Rightarrow\) 280	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.
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.
88:	\(C(\infty ,2)\): Every family of pairs has a choice function.
142:	\(\neg PB\): There is a set of reals without the property of Baire. Jech [1973b], p. 7.
280:	There is a complete separable metric space with a subset which does not have the Baire property.

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