Implications

This non-implication, Form 141 \( \not \Rightarrow \) Form 261, whose code is 4, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1388, whose string of implications is:
    345 \(\Rightarrow\) 14 \(\Rightarrow\) 141
  • A proven non-implication whose code is 3. In this case, it's Code 3: 266, Form 345 \( \not \Rightarrow \) Form 51 whose summary information is:
    Hypothesis Statement
    Form 345 <p> <strong>Rasiowa-Sikorski Axiom:</strong>  If \((B,\land,\lor)\) is a Boolean algebra, \(a\) is a non-zero element of \(B\), and \(\{X_n: n\in\omega\}\) is a denumerable set of subsets of \(B\) then there is a maximal filter \(F\) of \(B\) such that \(a\in F\) and for each \(n\in\omega\), if \(X_n\subseteq F\) and \(\bigwedge X_n\) exists then \(\bigwedge X_n \in F\). </p>

    Conclusion Statement
    Form 51 <p> <strong>Cofinality Principle:</strong> Every linear ordering has a cofinal sub well ordering.  <a href="/articles/Sierpi\'nski-1918">Sierpi\'nski [1918]</a>, p 117. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8650, whose string of implications is:
    261 \(\Rightarrow\) 256 \(\Rightarrow\) 259 \(\Rightarrow\) 51

The conclusion Form 141 \( \not \Rightarrow \) Form 261 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal M40(\kappa)\) Pincus' Model IV The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\)
\(\cal N40\) Howard/Rubin Model II A variation of \(\cal N38\)

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