Implications

This non-implication, Form 39 \( \not \Rightarrow \) Form 109, 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: 110, Form 39 \( \not \Rightarrow \) Form 106 whose summary information is:
    Hypothesis Statement
    Form 39 <p> \(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. <a href="/books/2">Moore, G. [1982]</a>, p. 202. </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: 4909, whose string of implications is:
    109 \(\Rightarrow\) 66 \(\Rightarrow\) 67 \(\Rightarrow\) 106

The conclusion Form 39 \( \not \Rightarrow \) Form 109 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

Edit | Back