Implications

This non-implication, Form 199(\(n\)) \( \not \Rightarrow \) Form 28-p, 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: 2680, whose string of implications is:
    40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 13 \(\Rightarrow\) 199(\(n\))
  • A proven non-implication whose code is 3. In this case, it's Code 3: 214, Form 40 \( \not \Rightarrow \) Form 222 whose summary information is:
    Hypothesis Statement
    Form 40 <p> \(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. <a href="/books/2">Moore, G. [1982]</a>, p 325. </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: 5145, whose string of implications is:
    28-p \(\Rightarrow\) 427 \(\Rightarrow\) 67 \(\Rightarrow\) 52 \(\Rightarrow\) 221 \(\Rightarrow\) 222

The conclusion Form 199(\(n\)) \( \not \Rightarrow \) Form 28-p 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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