Implications

This non-implication, Form 189 \( \not \Rightarrow \) Form 183-alpha, whose code is 6, 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: 9615, whose string of implications is:
    191 \(\Rightarrow\) 189
  • A proven non-implication whose code is 5. In this case, it's Code 3: 488, Form 191 \( \not \Rightarrow \) Form 183-alpha whose summary information is:
    Hypothesis Statement
    Form 191 <p> \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). </p>

    Conclusion Statement
    Form 183-alpha <p> There are no \(\aleph_{\alpha}\) minimal  sets.  That is, there are no sets \(X\) such that <ol type="1"> <li>\(|X|\) is incomparable with \(\aleph_{\alpha}\)</li> <li>\(\aleph_{\beta}<|X|\) for every \(\beta < \alpha \) and</li> <li>\(\forall Y\subseteq X, |Y|<\aleph_{\alpha}\) or \(|X-Y| <\aleph_{\alpha}\).</li> </ol> </p>

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

The conclusion Form 189 \( \not \Rightarrow \) Form 183-alpha then follows.

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

Name Statement
\(\cal N27\) Hickman's Model II Let \(A\) be a set with cardinality\(\aleph_1\) such that \(A=\{(a_{\alpha},b_{\beta}) : \alpha < \omega, \beta< \omega_1\}\)

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