Implications

This non-implication, Form 122 \( \not \Rightarrow \) Form 301, 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: 10202, whose string of implications is:
    165 \(\Rightarrow\) 122
  • 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: 9606, whose string of implications is:
    301 \(\Rightarrow\) 300

The conclusion Form 122 \( \not \Rightarrow \) Form 301 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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