Implications

This non-implication, Form 142 \( \not \Rightarrow \) Form 311, 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: 212, Form 142 \( \not \Rightarrow \) Form 222 whose summary information is:
    Hypothesis 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>

    Conclusion Statement
    Form 222 <p> There is a non-principal measure on \(\cal P(\omega)\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8932, whose string of implications is:
    311 \(\Rightarrow\) 313 \(\Rightarrow\) 222

The conclusion Form 142 \( \not \Rightarrow \) Form 311 then follows.

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

Name Statement
\(\cal M2\) Feferman's model Add a denumerable number of generic reals to the base model, but do not collect them

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