Implications

We have the following indirect implication of form equivalence classes:

326 \(\Rightarrow\) 18
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
326 \(\Rightarrow\) 88 Logic at Work: Essay Dedicated to the Memory of Helen Rasiowa, Wojtylak, 1999,
88 \(\Rightarrow\) 80 clear
80 \(\Rightarrow\) 18 clear

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

Howard-Rubin Number Statement
326:

2-SAT:  Restricted Compactness Theorem for Propositional Logic III:   If \(\Sigma\) is a set of formulas in a propositional language such that every finite subset of \(\Sigma\) is satisfiable and if every formula in \(\Sigma\) is a disjunction of at most two literals, then \(\Sigma\) is satisfiable. (A literal is a propositional variable or its negation.) Wojtylak [1999] (listed as Wojtylak [1995])

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.

18:

\(PUT(\aleph_{0},2,\aleph_{0})\):  The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.

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