Axiom Of Choice

We have the following indirect implication of form equivalence classes:

213 \(\Rightarrow\) 280 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
213 \(\Rightarrow\) 85	clear
85 \(\Rightarrow\) 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
213:	\(C(\infty,\aleph_{1})\): If \((\forall y\in X)(\|y\| = \aleph_{1})\) then \(X\) has a choice function.
85:	\(C(\infty,\aleph_{0})\): Every family of denumerable sets has a choice function. Jech [1973b] p 115 prob 7.13.
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.

Comment:

Back