Implications

This non-implication, Form 332 \( \not \Rightarrow \) Form 302, 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: 1618, whose string of implications is:
    14 \(\Rightarrow\) 123 \(\Rightarrow\) 331 \(\Rightarrow\) 332
  • A proven non-implication whose code is 3. In this case, it's Code 3: 3, Form 14 \( \not \Rightarrow \) Form 300 whose summary information is:
    Hypothesis Statement
    Form 14 <p> <strong>BPI:</strong> Every Boolean algebra has a prime ideal. </p>

    Conclusion Statement
    Form 300 <p> Any continuous surjection between extremally disconnected compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8890, whose string of implications is:
    302 \(\Rightarrow\) 301 \(\Rightarrow\) 300

The conclusion Form 332 \( \not \Rightarrow \) Form 302 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal M1\) Cohen's original model Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them

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