Implications

This non-implication, Form 65 \( \not \Rightarrow \) Form 174-alpha, whose code is 6, is constructed around a proven non-implication as follows:

  • 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: 139, Form 65 \( \not \Rightarrow \) Form 18 whose summary information is:
    Hypothesis Statement
    Form 65 <p> <strong>The Krein-Milman Theorem:</strong> Let \(K\) be a compact convex set in a locally convex topological vector space \(X\). Then \(K\) has an extreme point. (An <em>extreme point</em> is a point which is not an interior point of any line segment which lies in  \(K\).) <a href="/books/23">Rubin, H./Rubin, J. [1985]</a> p. 177. <p>

    Conclusion Statement
    Form 18 <p> \(PUT(\aleph_{0},2,\aleph_{0})\):  The union of a denumerable family of pairwise disjoint pairs has a denumerable subset. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1039, whose string of implications is:
    174-alpha \(\Rightarrow\) 9 \(\Rightarrow\) 17 \(\Rightarrow\) 18

The conclusion Form 65 \( \not \Rightarrow \) Form 174-alpha then follows.

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

Name Statement
\(\cal N2\) The Second Fraenkel Model The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \).

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