Implications

This non-implication, Form 38 \( \not \Rightarrow \) Form 270, 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: 1889, whose string of implications is:
    23 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 6 \(\Rightarrow\) 5 \(\Rightarrow\) 38
  • A proven non-implication whose code is 5. In this case, it's Code 3: 50, Form 23 \( \not \Rightarrow \) Form 127 whose summary information is:
    Hypothesis Statement
    Form 23 <p> \((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\). </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: 4560, whose string of implications is:
    270 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 45-n \(\Rightarrow\) 64 \(\Rightarrow\) 127

The conclusion Form 38 \( \not \Rightarrow \) Form 270 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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