Implications

This non-implication, Form 65 \( \not \Rightarrow \) Form 323, 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: 4470, whose string of implications is:
    323 \(\Rightarrow\) 62 \(\Rightarrow\) 10 \(\Rightarrow\) 80 \(\Rightarrow\) 18

The conclusion Form 65 \( \not \Rightarrow \) Form 323 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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