Implications

This non-implication, Form 42 \( \not \Rightarrow \) Form 50, 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: 1424, Form 42 \( \not \Rightarrow \) Form 371 whose summary information is:
    Hypothesis Statement
    Form 42 <p> <strong>L&ouml;wenheim-Skolem Theorem:</strong> If a countable family of first order  sentences is satisfiable in a set \(M\) then it is satisfiable in a countable subset of \(M\). (See <a href="/books/2">Moore, G. [1982]</a>, p. 251 for references. </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: 1369, whose string of implications is:
    50 \(\Rightarrow\) 14 \(\Rightarrow\) 371

The conclusion Form 42 \( \not \Rightarrow \) Form 50 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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