Implications

This non-implication, Form 57 \( \not \Rightarrow \) Form 264, 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: 3703, whose string of implications is:
    345 \(\Rightarrow\) 43 \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 57
  • 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: 7861, whose string of implications is:
    264 \(\Rightarrow\) 202 \(\Rightarrow\) 181

The conclusion Form 57 \( \not \Rightarrow \) Form 264 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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