Implications

This non-implication, Form 119 \( \not \Rightarrow \) Form 41, 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: 7445, whose string of implications is:
    115 \(\Rightarrow\) 118 \(\Rightarrow\) 119
  • A proven non-implication whose code is 5. In this case, it's Code 3: 295, Form 115 \( \not \Rightarrow \) Form 128 whose summary information is:
    Hypothesis Statement
    Form 115 <p> The product of weakly Loeb \(T_2\) spaces is weakly Loeb. <em>Weakly Loeb</em> means the set of non-empty closed subsets has a multiple choice function.) </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: 890, whose string of implications is:
    41 \(\Rightarrow\) 9 \(\Rightarrow\) 128

The conclusion Form 119 \( \not \Rightarrow \) Form 41 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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