Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
295 \(\Rightarrow\) 30	"Dense orderings, partitions, and weak forms of choice", Gonzalez, C. 1995a, Fund. Math.
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

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

Howard-Rubin Number	Statement
295:	DO: Every infinite set has a dense linear ordering.
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.

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