Axiom Of Choice

We have the following indirect implication of form equivalence classes:

391 \(\Rightarrow\) 362 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
391 \(\Rightarrow\) 112	clear
112 \(\Rightarrow\) 90	Equivalents of the Axiom of Choice II, Rubin/Rubin, 1985, page 79
90 \(\Rightarrow\) 91	The Axiom of Choice, Jech, 1973b, page 133
91 \(\Rightarrow\) 361	Equivalents of the Axiom of Choice II, Rubin, 1985, theorem 5.7
361 \(\Rightarrow\) 362	Zermelo's Axiom of Choice, Moore, 1982, page 325

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

Howard-Rubin Number	Statement
391:	\(C(\infty,LO)\): Every set of non-empty linearly orderable sets has a choice function.
112:	\(MC(\infty,LO)\): For every family \(X\) of non-empty sets each of which can be linearly ordered there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
90:	\(LW\): Every linearly ordered set can be well ordered. Jech [1973b], p 133.
91:	\(PW\): The power set of a well ordered set can be well ordered.
361:	In \(\Bbb R\), the union of a denumerable number of analytic sets is analytic. G. Moore [1982], pp 181 and 325.
362:	In \(\Bbb R\), every Borel set is analytic. G. Moore [1982], pp 181 and 325.

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