Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 286, whose code is 4, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 11140, whose string of implications is:
    253 \(\Rightarrow\) 0
  • A proven non-implication whose code is 3. In this case, it's Code 3: 279, Form 253 \( \not \Rightarrow \) Form 286 whose summary information is:
    Hypothesis Statement
    Form 253 <p> <strong>\L o\'s' Theorem:</strong> If \(M=\langle A,R_j\rangle_{j\in J}\) is a relational system, \(X\) any set and \({\cal F}\) an ultrafilter in \({\cal P}(X)\), then \(M\) and \(M^{X}/{\cal F}\) are elementarily equivalent. </p>

    Conclusion Statement
    Form 286 <p> <strong>Extended Krein-Milman Theorem:</strong>  Let K be a quasicompact (sometimes called convex-compact), convex subset of a locally convex topological vector space, then K has an extreme point. <a href="/books/10">H. Rubin/J. Rubin [1985]</a>, p. 177-178. </p>

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

The conclusion Form 0 \( \not \Rightarrow \) Form 286 then follows.

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

Name Statement
\(\cal M27\) Pincus/Solovay Model I Let \(\cal M_1\) be a model of \(ZFC + V =L\)

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