Implications

This non-implication, Form 111 \( \not \Rightarrow \) Form 331, 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: 9100, whose string of implications is:
    331 \(\Rightarrow\) 332 \(\Rightarrow\) 343 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 45-n \(\Rightarrow\) 64 \(\Rightarrow\) 127

The conclusion Form 111 \( \not \Rightarrow \) Form 331 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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