Implications

This non-implication, Form 169 \( \not \Rightarrow \) Form 306, 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: 240, Form 169 \( \not \Rightarrow \) Form 93 whose summary information is:
    Hypothesis Statement
    Form 169 <p> There is an uncountable subset of \({\Bbb R}\) without a perfect subset. </p>

    Conclusion Statement
    Form 93 <p> There is a non-measurable subset of \({\Bbb R}\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9874, whose string of implications is:
    306 \(\Rightarrow\) 93

The conclusion Form 169 \( \not \Rightarrow \) Form 306 then follows.

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

Name Statement
\(\cal M38\) Shelah's Model II In a model of \(ZFC +\) "\(\kappa\) is a strongly inaccessible cardinal", Shelah uses Levy's method of collapsing cardinals to collapse \(\kappa\) to \(\aleph_1\) similarly to <a href="/articles/Solovay-1970">Solovay [1970]</a>

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