Implications

This non-implication, Form 183-alpha \( \not \Rightarrow \) Form 426, whose code is 4, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10308, whose string of implications is:
    71-alpha \(\Rightarrow\) 183-alpha
  • A proven non-implication whose code is 3. In this case, it's Code 3: 129, Form 71-alpha \( \not \Rightarrow \) Form 8 whose summary information is:
    Hypothesis Statement
    Form 71-alpha  <p> \(W_{\aleph_{\alpha}}\): \((\forall x)(|x|\le\aleph_{\alpha }\) or \(|x|\ge \aleph_{\alpha})\). <a href="/books/8">Jech [1973b]</a>, page 119. </p>

    Conclusion Statement
    Form 8 <p> \(C(\aleph_{0},\infty)\): </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9683, whose string of implications is:
    426 \(\Rightarrow\) 8

The conclusion Form 183-alpha \( \not \Rightarrow \) Form 426 then follows.

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

Name Statement
\(\cal N16(\aleph_{\gamma})\) Levy's Model II This is an extension of\(\cal N16\) in which \(A\) has cardinality \(\aleph_{\gamma}\) wherecf\((\aleph_{\gamma}) = \aleph_0\); \(\cal G\) is the group of allpermutations on \(A\); and \(S\) is the set of all subsets of \(A\) ofcardinality less that \(\aleph_{\gamma}\)

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