Implications

This non-implication, Form 124 \( \not \Rightarrow \) Form 257, 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: 1028, whose string of implications is:
    41 \(\Rightarrow\) 9 \(\Rightarrow\) 17 \(\Rightarrow\) 124
  • 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: 8752, whose string of implications is:
    257 \(\Rightarrow\) 260 \(\Rightarrow\) 40 \(\Rightarrow\) 43 \(\Rightarrow\) 106

The conclusion Form 124 \( \not \Rightarrow \) Form 257 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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