Implications

This non-implication, Form 30 \( \not \Rightarrow \) 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: 9682, whose string of implications is:
    295 \(\Rightarrow\) 30
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1441, Form 295 \( \not \Rightarrow \) Form 384 whose summary information is:
    Hypothesis Statement
    Form 295 <p> <strong>DO:</strong>  Every infinite set has a dense linear ordering. </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 30 \( \not \Rightarrow \) Form 384 then follows.

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

Name Statement
\(\cal M3\) Mathias' model Mathias proves that the \(FM\) model <a href="/models/Mathias-Pincus-1">\(\cal N4\)</a> can be transformed into a model of \(ZF\), \(\cal M3\)
\(\cal M40(\kappa)\) Pincus' Model IV The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\)
\(\cal M45\) Pincus' Model VII This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((C)\)

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