Implications

This non-implication, Form 88 \( \not \Rightarrow \) Form 384, 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: 1582, whose string of implications is:
    14 \(\Rightarrow\) 49 \(\Rightarrow\) 201 \(\Rightarrow\) 88
  • A proven non-implication whose code is 3. In this case, it's Code 3: 96, Form 14 \( \not \Rightarrow \) Form 384 whose summary information is:
    Hypothesis Statement
    Form 14 <p> <strong>BPI:</strong> Every Boolean algebra has a prime ideal. </p>

    Conclusion Statement
    Form 384 <p> <strong>Closed Filter Extendability for \(T_1\) Spaces</strong>: Every closed filter in a \(T_1\) topological space can be extended to a maximal closed filter. </p>

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

The conclusion Form 88 \( \not \Rightarrow \) Form 384 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 \(AC\)

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