Implications

This non-implication, Form 5 \( \not \Rightarrow \) Form 363, 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: 9875, whose string of implications is:
    6 \(\Rightarrow\) 5
  • A proven non-implication whose code is 3. In this case, it's Code 3: 248, Form 6 \( \not \Rightarrow \) Form 273 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 273 <p> There is a subset of \({\Bbb R}\) which is not Borel. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9365, whose string of implications is:
    363 \(\Rightarrow\) 364 \(\Rightarrow\) 273

The conclusion Form 5 \( \not \Rightarrow \) Form 363 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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