Implications

This non-implication, Form 204 \( \not \Rightarrow \) Form 214, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1087, Form 204 \( \not \Rightarrow \) Form 152 whose summary information is:
    Hypothesis Statement
    Form 204 <p> For every infinite \(X\), there is a function from \(X\) onto \(2X\). </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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10151, whose string of implications is:
    214 \(\Rightarrow\) 152

The conclusion Form 204 \( \not \Rightarrow \) Form 214 then follows.

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

Name Statement
\(\cal M2\) Feferman's model Add a denumerable number of generic reals to the base model, but do not collect them

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