Implications

This non-implication, Form 169 \( \not \Rightarrow \) Form 334, 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: 5106, whose string of implications is:
    334 \(\Rightarrow\) 67 \(\Rightarrow\) 52 \(\Rightarrow\) 93

The conclusion Form 169 \( \not \Rightarrow \) Form 334 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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