Implications

This non-implication, Form 170 \( \not \Rightarrow \) Form 418, 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: 76, Form 170 \( \not \Rightarrow \) Form 34 whose summary information is:
    Hypothesis Statement
    Form 170 <p> \(\aleph_{1}\le 2^{\aleph_{0}}\). </p>

    Conclusion Statement
    Form 34 <p> \(\aleph_{1}\) is regular. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9582, whose string of implications is:
    418 \(\Rightarrow\) 419 \(\Rightarrow\) 420 \(\Rightarrow\) 34

The conclusion Form 170 \( \not \Rightarrow \) Form 418 then follows.

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

Name Statement
\(\cal M36\) Figura's Model Starting with a countable, standard model, \(\cal M\), of \(ZFC + 2^{\aleph_0}=\aleph_{\omega +1}\), Figura uses forcing conditions that are functions from a subset of \(\omega\times\omega\) to \(\omega_\omega\) to construct a symmetric extension of \(\cal M\) in which there is an uncountable well ordered subset of the reals (<a href="/form-classes/howard-rubin-170">Form 170</a> is true), but \(\aleph_1= \aleph_{\omega}\) so \(\aleph_1\) is singular (<a href="/form-classes/howard-rubin-34">Form 34</a> is false)

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