Implications

This non-implication, Form 43 \( \not \Rightarrow \) Form 256, 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: 10295, whose string of implications is:
    87-alpha \(\Rightarrow\) 43
  • A proven non-implication whose code is 3. In this case, it's Code 3: 896, Form 87-alpha \( \not \Rightarrow \) Form 51 whose summary information is:
    Hypothesis Statement
    Form 87-alpha <p> \(DC(\aleph_{\alpha})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(|Y|<\aleph_{\alpha}\), there is an \(x\in X\) with \(Y\mathrel R x\) then there is a function \(f:\aleph_{\alpha}\to X\) such that (\(\forall\beta < \aleph_{\alpha}\)) \(\{f(\gamma): \gamma < \beta\}\mathrel R f(\beta)\). </p>

    Conclusion Statement
    Form 51 <p> <strong>Cofinality Principle:</strong> Every linear ordering has a cofinal sub well ordering.  <a href="/articles/Sierpi\'nski-1918">Sierpi\'nski [1918]</a>, p 117. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8646, whose string of implications is:
    256 \(\Rightarrow\) 259 \(\Rightarrow\) 51

The conclusion Form 43 \( \not \Rightarrow \) Form 256 then follows.

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

Name Statement
\(\cal M40(\kappa)\) Pincus' Model IV The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\)

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