Implications

This non-implication, Form 324 \( \not \Rightarrow \) Form 311, 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: 9966, whose string of implications is:
    165 \(\Rightarrow\) 324
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1175, Form 165 \( \not \Rightarrow \) Form 222 whose summary information is:
    Hypothesis Statement
    Form 165 <p> \(C(WO,WO)\):  Every well ordered family of non-empty, well orderable sets has a choice function. </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 324 \( \not \Rightarrow \) 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

Edit | Back