Implications

This non-implication, Form 280 \( \not \Rightarrow \) Form 222, whose code is 4, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10287, whose string of implications is:
    142 \(\Rightarrow\) 280
  • 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>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 280 \( \not \Rightarrow \) Form 222 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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