Implications

This non-implication, Form 93 \( \not \Rightarrow \) Form 381, 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: 10172, whose string of implications is:
    170 \(\Rightarrow\) 93
  • 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: 9412, whose string of implications is:
    381 \(\Rightarrow\) 418 \(\Rightarrow\) 419 \(\Rightarrow\) 420 \(\Rightarrow\) 34

The conclusion Form 93 \( \not \Rightarrow \) Form 381 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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