Implications

This non-implication, Form 342-n \( \not \Rightarrow \) Form 152, 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: 3801, whose string of implications is:
    87-alpha \(\Rightarrow\) 43 \(\Rightarrow\) 8 \(\Rightarrow\) 342-n
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1086, Form 87-alpha \( \not \Rightarrow \) Form 152 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 152 <p> \(D_{\aleph_{0}}\): Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets.  (See <a href=""notes/note-27">note 27</a> for \(D_{\kappa}\), \(\kappa\) a well ordered cardinal.) </p>

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

The conclusion Form 342-n \( \not \Rightarrow \) Form 152 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 \(AC\)

Edit | Back