Implications

This non-implication, Form 165 \( \not \Rightarrow \) Form 302, 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: 1272, Form 165 \( \not \Rightarrow \) Form 300 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 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 165 \( \not \Rightarrow \) 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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