Implications

We have the following indirect implication of form equivalence classes:

326 \(\Rightarrow\) 276
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 \(\Rightarrow\) 276

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.

276:

\(E(V'',III)\): For every set \(A\), \({\cal P}(A)\) is Dedekind finite if and only if \(A = \emptyset\)  or \(2|{\cal P}(A)| > |{\cal P}(A)|\). \ac{Howard/Spi\u siak} \cite{1994}.

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