Implications

This non-implication, Form 204 \( \not \Rightarrow \) Form 14, 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: 255, Form 204 \( \not \Rightarrow \) Form 70 whose summary information is:
    Hypothesis Statement
    Form 204 <p> For every infinite \(X\), there is a function from \(X\) onto \(2X\). </p>

    Conclusion Statement
    Form 70 <p> There is a non-trivial ultrafilter on \(\omega\). <a href="/books/8">Jech [1973b]</a>, prob 5.24. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1456, whose string of implications is:
    14 \(\Rightarrow\) 63 \(\Rightarrow\) 70

The conclusion Form 204 \( \not \Rightarrow \) Form 14 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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