Implications

This non-implication, Form 369 \( \not \Rightarrow \) Form 292, 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: 1192, Form 369 \( \not \Rightarrow \) Form 273 whose summary information is:
    Hypothesis Statement
    Form 369 <p> If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\). </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: 6702, whose string of implications is:
    292 \(\Rightarrow\) 90 \(\Rightarrow\) 91 \(\Rightarrow\) 273

The conclusion Form 369 \( \not \Rightarrow \) Form 292 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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