Implications

This non-implication, Form 38 \( \not \Rightarrow \) Form 419, 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: 221, whose string of implications is:
    6 \(\Rightarrow\) 5 \(\Rightarrow\) 38
  • A proven non-implication whose code is 3. In this case, it's Code 3: 879, Form 6 \( \not \Rightarrow \) Form 34 whose summary information is:
    Hypothesis Statement
    Form 6 <p> \(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable  family  of denumerable subsets of \({\Bbb R}\) is denumerable. </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: 9588, whose string of implications is:
    419 \(\Rightarrow\) 420 \(\Rightarrow\) 34

The conclusion Form 38 \( \not \Rightarrow \) Form 419 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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