Implications

This non-implication, Form 128 \( \not \Rightarrow \) Form 292, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1230, Form 128 \( \not \Rightarrow \) Form 289 whose summary information is:
    Hypothesis Statement
    Form 128 <p> <strong>Aczel's Realization Principle:</strong> On every infinite set there is a Hausdorff topology with an infinite set of non-isolated points. </p>

    Conclusion Statement
    Form 289 <p> If \(S\) is a set of subsets of a countable set and \(S\) is closed under chain unions, then \(S\) has a \(\subseteq\)-maximal element. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6725, whose string of implications is:
    292 \(\Rightarrow\) 90 \(\Rightarrow\) 91 \(\Rightarrow\) 79 \(\Rightarrow\) 289

The conclusion Form 128 \( \not \Rightarrow \) Form 292 then follows.

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

Name Statement
\(\cal M1\) Cohen's original model Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them

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