Axiom Of Choice

We have the following indirect implication of form equivalence classes:

61 \(\Rightarrow\) 142 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
61 \(\Rightarrow\) 88	clear
88 \(\Rightarrow\) 142	The Axiom of Choice, Jech, 1973b, page 7 problem 11

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

Howard-Rubin Number	Statement
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.

Comment:

Back