Implications

This non-implication, Form 93 \( \not \Rightarrow \) Form 385, 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: 5891, whose string of implications is:
    385 \(\Rightarrow\) 70 \(\Rightarrow\) 142

The conclusion Form 93 \( \not \Rightarrow \) Form 385 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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