Implications

This non-implication, Form 355 \( \not \Rightarrow \) Form 328, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 3685, whose string of implications is:
    345 \(\Rightarrow\) 43 \(\Rightarrow\) 8 \(\Rightarrow\) 355
  • A proven non-implication whose code is 5. In this case, it's Code 3: 710, Form 345 \( \not \Rightarrow \) Form 328 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 328 <p> \(MC(WO,\infty)\): For  every well ordered set \(X\) such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that and for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). (See <a href="/form-classes/howard-rubin-67">Form 67</a>.) </p>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 355 \( \not \Rightarrow \) Form 328 then follows.

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

Name Statement
\(\cal N40\) Howard/Rubin Model II A variation of \(\cal N38\)

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