Implications

We have the following indirect implication of form equivalence classes:

201 \(\Rightarrow\) 389
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
201 \(\Rightarrow\) 88 The dependence of some logical axioms on disjoint transversals and linked systems, Schrijver, A. 1978, Colloq. Math.
88 \(\Rightarrow\) 80 clear
80 \(\Rightarrow\) 389 clear

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

Howard-Rubin Number Statement
201:

Linking Axiom for Boolean Algebras: Every Boolean algebra has a maximal linked system. (\(L\subseteq B\) is linked if \(a\wedge b\neq 0\) for all \(a\) and \(b \in L\).)

88:

  \(C(\infty ,2)\):  Every family of pairs has a choice function.

80:

\(C(\aleph_{0},2)\):  Every denumerable set of  pairs has  a  choice function.

389:

\(C(\aleph_0,2,\cal P({\Bbb R}))\): Every denumerable family of two element subsets of \(\cal P({\Bbb R})\) has a choice function.  \ac{Keremedis} \cite{1999b}.

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