Implications

This non-implication, Form 365 \( \not \Rightarrow \) Form 430-p, whose code is 6, 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: 10139, whose string of implications is:
    41 \(\Rightarrow\) 365
  • A proven non-implication whose code is 5. In this case, it's Code 3: 113, Form 41 \( \not \Rightarrow \) Form 106 whose summary information is:
    Hypothesis Statement
    Form 41 <p> \(W_{\aleph _{1}}\): For every cardinal \(m\), \(m \le \aleph_{1}\) or \(\aleph_{1}\le m \). </p>

    Conclusion Statement
    Form 106 <p> <strong>Baire Category Theorem for Compact Hausdorff Spaces:</strong> Every compact Hausdorff space is Baire. <p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5211, whose string of implications is:
    430-p \(\Rightarrow\) 67 \(\Rightarrow\) 106

The conclusion Form 365 \( \not \Rightarrow \) Form 430-p then follows.

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

Name Statement
\(\cal N16\) Jech/Levy/Pincus Model \(A\) has cardinality \(\aleph_{\omega}\);\(\cal G\) is the group of all permutations on \(A\); and \(S\) is the set ofall subsets of \(A\) of cardinality less that \(\aleph_{\omega}\)

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