Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 169, 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: 11372, whose string of implications is:
    369 \(\Rightarrow\) 0
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1121, Form 369 \( \not \Rightarrow \) Form 169 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 169 <p> There is an uncountable subset of \({\Bbb R}\) without a perfect subset. </p>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 0 \( \not \Rightarrow \) Form 169 then follows.

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

Name Statement
\(\cal M5(\aleph)\) Solovay's Model An inaccessible cardinal \(\aleph\) is collapsed to \(\aleph_1\) in the outer model and then \(\cal M5(\aleph)\) is the smallest model containing the ordinals and \(\Bbb R\)
\(\cal M12(\aleph)\) 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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