Implications

This non-implication, Form 119 \( \not \Rightarrow \) Form 258, 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: 2450, whose string of implications is:
    231 \(\Rightarrow\) 32 \(\Rightarrow\) 119
  • A proven non-implication whose code is 3. In this case, it's Code 3: 880, Form 231 \( \not \Rightarrow \) Form 34 whose summary information is:
    Hypothesis Statement
    Form 231 <p> \(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets is well orderable. </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: 8327, whose string of implications is:
    258 \(\Rightarrow\) 255 \(\Rightarrow\) 260 \(\Rightarrow\) 40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 34

The conclusion Form 119 \( \not \Rightarrow \) Form 258 then follows.

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

Name Statement
\(\cal M12(\aleph)\) Truss' Model I This is a variation of Solovay's model, <a href="/models/Solovay-1">\(\cal M5(\aleph)\)</a> in which \(\aleph\) is singular

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