Implications

This non-implication, Form 106 \( \not \Rightarrow \) Form 152, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 3902, whose string of implications is:
    44 \(\Rightarrow\) 43 \(\Rightarrow\) 106
  • A proven non-implication whose code is 5. In this case, it's Code 3: 115, Form 44 \( \not \Rightarrow \) Form 152 whose summary information is:
    Hypothesis Statement
    Form 44 <p> \(DC(\aleph _{1})\):  Given a relation \(R\) such that for every  subset \(Y\) of a set \(X\) with \(|Y| < \aleph_{1}\) there is an \(x \in  X\)  with \(Y \mathrel R x\), then there is a function \(f: \aleph_{1} \rightarrow  X\) such that \((\forall\beta < \aleph_{1}) (\{f(\gamma ): \gamma < b \} \mathrel R f(\beta))\). </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 106 \( \not \Rightarrow \) Form 152 then follows.

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

Name Statement
\(\cal N12(\aleph_2)\) Another variation of \(\cal N1\) Change "\(\aleph_1\)" to "\(\aleph_2\)" in \(\cal N12(\aleph_1)\) above

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