Implications

We have the following indirect implication of form equivalence classes:

201 \(\Rightarrow\) 142
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 \(\Rightarrow\) 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
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.

142:

\(\neg  PB\):  There is a set of reals without the property of Baire.  Jech [1973b], p. 7.

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