Implications

This non-implication, Form 92 \( \not \Rightarrow \) Form 128, 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: 10235, whose string of implications is:
    337 \(\Rightarrow\) 92
  • A proven non-implication whose code is 5. In this case, it's Code 3: 687, Form 337 \( \not \Rightarrow \) Form 128 whose summary information is:
    Hypothesis Statement
    Form 337 <p> \(C(WO\), <strong>uniformly linearly ordered</strong>):  If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\). </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>

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

The conclusion Form 92 \( \not \Rightarrow \) Form 128 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 \(A\)
\(\cal N2\) The Second Fraenkel Model The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \).

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