Implications

This non-implication, Form 111 \( \not \Rightarrow \) Form 53, 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: 7514, whose string of implications is:
    23 \(\Rightarrow\) 151 \(\Rightarrow\) 122 \(\Rightarrow\) 250 \(\Rightarrow\) 111
  • A proven non-implication whose code is 5. In this case, it's Code 3: 37, Form 23 \( \not \Rightarrow \) Form 53 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 53 <p> For all infinite cardinals \(m\), \(m^2\le 2^m\).  <a href="/articles/Mathias-1979">Mathias [1979]</a>, prob 1336. </p>

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

The conclusion Form 111 \( \not \Rightarrow \) Form 53 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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