Implications

This non-implication, Form 253 \( \not \Rightarrow \) Form 264, whose code is 4, 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 3. In this case, it's Code 3: 1426, Form 253 \( \not \Rightarrow \) Form 371 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 371 <p> There is an infinite, compact, Hausdorff, extremally disconnected topological space.  <a href="/excerpts/Morillon-1993-1">Morillon [1993]</a>. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7837, whose string of implications is:
    264 \(\Rightarrow\) 202 \(\Rightarrow\) 91 \(\Rightarrow\) 79 \(\Rightarrow\) 371

The conclusion Form 253 \( \not \Rightarrow \) Form 264 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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