Implications

This non-implication, Form 71-alpha \( \not \Rightarrow \) Form 174-alpha, 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)

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 5. In this case, it's Code 3: 166, Form 71-alpha \( \not \Rightarrow \) Form 106 whose summary information is:
    Hypothesis Statement
    Form 71-alpha  <p> \(W_{\aleph_{\alpha}}\): \((\forall x)(|x|\le\aleph_{\alpha }\) or \(|x|\ge \aleph_{\alpha})\). <a href="/books/8">Jech [1973b]</a>, page 119. </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: 3890, whose string of implications is:
    174-alpha \(\Rightarrow\) 43 \(\Rightarrow\) 106

The conclusion Form 71-alpha \( \not \Rightarrow \) Form 174-alpha then follows.

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

Name Statement
\(\cal N16(\aleph_{\gamma})\) Levy's Model II This is an extension of\(\cal N16\) in which \(A\) has cardinality \(\aleph_{\gamma}\) wherecf\((\aleph_{\gamma}) = \aleph_0\); \(\cal G\) is the group of allpermutations on \(A\); and \(S\) is the set of all subsets of \(A\) ofcardinality less that \(\aleph_{\gamma}\)
\(\cal N21(\aleph_{\alpha+1})\) Jensen's Model We assume \(\aleph_{\alpha+1}\) is a regular cardinal

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