Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 25, 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: 81, Form 0 \( \not \Rightarrow \) Form 315 whose summary information is:
    Hypothesis Statement
    Form 0  \(0 = 0\).

    Conclusion Statement
    Form 315 <p> \(\Omega = \omega_1\), where<br />\(\Omega = \{\alpha\in\hbox{ On}: (\forall\beta\le\alpha)(\beta=0 \vee (\exists\gamma)(\beta=\gamma+1) \vee\)<br /> there is a sequence \(\langle\gamma_n: n\in\omega\rangle\) such that for each \(n\),<br /> \(\gamma_n<\beta\hbox{ and } \beta=\bigcup_{n<\omega}\gamma_n.)\} \) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9958, whose string of implications is:
    25 \(\Rightarrow\) 315

The conclusion Form 0 \( \not \Rightarrow \) Form 25 then follows.

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

Name Statement
\(\cal M17\) Gitik's Model Using the assumption that for every ordinal \(\alpha\) there is a strongly compact cardinal \(\kappa\) such that \(\kappa >\alpha\), Gitik extends the universe \(V\) by a filter \(G\) generic over a proper class of forcing conditions

Edit | Back