Implications

This non-implication, Form 93 \( \not \Rightarrow \) Form 99, 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: 244, Form 93 \( \not \Rightarrow \) Form 142 whose summary information is:
    Hypothesis Statement
    Form 93 <p> There is a non-measurable subset of \({\Bbb R}\). </p>

    Conclusion Statement
    Form 142 <p> \(\neg  PB\):  There is a set of reals without the property of Baire.  <a href="/books/8">Jech [1973b]</a>, p. 7. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5879, whose string of implications is:
    99 \(\Rightarrow\) 70 \(\Rightarrow\) 142

The conclusion Form 93 \( \not \Rightarrow \) Form 99 then follows.

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

Name Statement
\(\cal M18\) Shelah's Model I Shelah modified Solovay's model, <a href="/models/Solovay-1">\(\cal M5\)</a>, and constructed a model without using an inaccessible cardinal in which the <strong>Principle of Dependent Choices</strong> (<a href="/form-classes/howard-rubin-43">Form 43</a>) is true and every set of reals has the property of Baire (<a href="/form-classes/howard-rubin-142">Form142</a> is false)

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