Implications

This non-implication, Form 389 \( \not \Rightarrow \) Form 52, 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: 2427, whose string of implications is:
    165 \(\Rightarrow\) 32 \(\Rightarrow\) 10 \(\Rightarrow\) 80 \(\Rightarrow\) 389
  • 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: 4195, whose string of implications is:
    52 \(\Rightarrow\) 221 \(\Rightarrow\) 222

The conclusion Form 389 \( \not \Rightarrow \) Form 52 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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