Implications

This non-implication, Form 196-alpha \( \not \Rightarrow \) Form 255, 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: 9647, whose string of implications is:
    86-alpha \(\Rightarrow\) 196-alpha
  • A proven non-implication whose code is 5. In this case, it's Code 3: 169, Form 86-alpha \( \not \Rightarrow \) Form 106 whose summary information is:
    Hypothesis Statement
    Form 86-alpha <p> \(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(|X| = \aleph_{\alpha }\), then \(X\) has a choice function. </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: 8106, whose string of implications is:
    255 \(\Rightarrow\) 260 \(\Rightarrow\) 40 \(\Rightarrow\) 43 \(\Rightarrow\) 106

The conclusion Form 196-alpha \( \not \Rightarrow \) Form 255 then follows.

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

Name Statement
\(\cal N21(\aleph_{\alpha+1})\) Jensen's Model We assume \(\aleph_{\alpha+1}\) is a regular cardinal

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