Implications

This non-implication, Form 5 \( \not \Rightarrow \) Form 399, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 230, whose string of implications is:
    16 \(\Rightarrow\) 6 \(\Rightarrow\) 5
  • A proven non-implication whose code is 5. In this case, it's Code 3: 16, Form 16 \( \not \Rightarrow \) Form 127 whose summary information is:
    Hypothesis Statement
    Form 16 <p> \(C(\aleph_{0},\le 2^{\aleph_{0}})\):  Every denumerable collection of non-empty sets  each with power \(\le  2^{\aleph_{0}}\) 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: 8983, whose string of implications is:
    399 \(\Rightarrow\) 323 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 11 \(\Rightarrow\) 12 \(\Rightarrow\) 336-n \(\Rightarrow\) 64 \(\Rightarrow\) 127

The conclusion Form 5 \( \not \Rightarrow \) Form 399 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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