Implications

This non-implication, Form 204 \( \not \Rightarrow \) Form 317, 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: 1459, whose string of implications is:
    317 \(\Rightarrow\) 14 \(\Rightarrow\) 63 \(\Rightarrow\) 70

The conclusion Form 204 \( \not \Rightarrow \) Form 317 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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