Implications

This non-implication, Form 119 \( \not \Rightarrow \) Form 259, 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: 1894, whose string of implications is:
    23 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 32 \(\Rightarrow\) 119
  • A proven non-implication whose code is 5. In this case, it's Code 3: 51, Form 23 \( \not \Rightarrow \) Form 128 whose summary information is:
    Hypothesis Statement
    Form 23 <p> \((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\). </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: 8804, whose string of implications is:
    259 \(\Rightarrow\) 260 \(\Rightarrow\) 40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 128

The conclusion Form 119 \( \not \Rightarrow \) Form 259 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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