Implications

This non-implication, Form 217 \( \not \Rightarrow \) Form 9, 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: 1028, Form 217 \( \not \Rightarrow \) Form 128 whose summary information is:
    Hypothesis Statement
    Form 217 <p> Every infinite partially ordered set has either an infinite chain or an infinite antichain. </p>

    Conclusion 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9641, whose string of implications is:
    9 \(\Rightarrow\) 128

The conclusion Form 217 \( \not \Rightarrow \) Form 9 then follows.

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

Name Statement
\(\cal N1\) The Basic Fraenkel Model The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)

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