Implications

This non-implication, Form 131 \( \not \Rightarrow \) Form 106, 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: 781, whose string of implications is:
    113 \(\Rightarrow\) 8 \(\Rightarrow\) 126 \(\Rightarrow\) 131
  • A proven non-implication whose code is 5. In this case, it's Code 3: 271, Form 113 \( \not \Rightarrow \) Form 106 whose summary information is:
    Hypothesis Statement
    Form 113 <p> <strong>Tychonoff's Compactness Theorem for Countably Many Spaces:</strong> The product of a countable set of compact spaces is compact. </p>

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

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 131 \( \not \Rightarrow \) Form 106 then follows.

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

Name Statement
\(\cal N38\) Howard/Rubin Model I Let \((A,\le)\) be an ordered set of atomswhich is order isomorphic to \({\Bbb Q}^\omega\), the set of all functionsfrom \(\omega\) into \(\Bbb Q\) ordered by the lexicographic ordering

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