Implications

This non-implication, Form 174-alpha \( \not \Rightarrow \) Form 51, 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: 9703, whose string of implications is:
    87-alpha \(\Rightarrow\) 174-alpha
  • 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>

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

The conclusion Form 174-alpha \( \not \Rightarrow \) Form 51 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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