Implications

This non-implication, Form 111 \( \not \Rightarrow \) Form 303, 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: 7481, whose string of implications is:
    121 \(\Rightarrow\) 122 \(\Rightarrow\) 250 \(\Rightarrow\) 111
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1013, Form 121 \( \not \Rightarrow \) Form 127 whose summary information is:
    Hypothesis Statement
    Form 121 <p> \(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function. </p>

    Conclusion Statement
    Form 127 <p> An amorphous power of a compact \(T_2\) space, which as a set is well orderable, is well orderable. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4121, whose string of implications is:
    303 \(\Rightarrow\) 50 \(\Rightarrow\) 14 \(\Rightarrow\) 49 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 64 \(\Rightarrow\) 127

The conclusion Form 111 \( \not \Rightarrow \) Form 303 then follows.

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

Name Statement
\(\cal N24\) Hickman's Model I This model is a variation of \(\cal N2\)

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