Implications

This non-implication, Form 376 \( \not \Rightarrow \) Form 202, 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: 3700, whose string of implications is:
    345 \(\Rightarrow\) 43 \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 376
  • A proven non-implication whose code is 5. In this case, it's Code 3: 707, Form 345 \( \not \Rightarrow \) Form 181 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 181 <p> \(C(2^{\aleph_0},\infty)\): Every set \(X\) of non-empty sets such that \(|X|=2^{\aleph_0}\) has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10209, whose string of implications is:
    202 \(\Rightarrow\) 181

The conclusion Form 376 \( \not \Rightarrow \) Form 202 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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