Implications

This non-implication, Form 253 \( \not \Rightarrow \) Form 99, 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: 14, Form 253 \( \not \Rightarrow \) Form 206 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 206 <p> The existence of a non-principal ultrafilter: There exists an infinite set \(X\) and a non-principal ultrafilter on \(X\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5881, whose string of implications is:
    99 \(\Rightarrow\) 70 \(\Rightarrow\) 206

The conclusion Form 253 \( \not \Rightarrow \) Form 99 then follows.

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

Name Statement
\(\cal M15\) Feferman/Blass Model Blass constructs a model similar to Feferman's model, <a href="/models/Feferman-1">\(\cal M2\)</a>
\(\cal M27\) Pincus/Solovay Model I Let \(\cal M_1\) be a model of \(ZFC + V =L\)
\(\cal M30\) Pincus/Solovay Model II In this construction, an \(\omega_1\) sequence of generic reals is added to a model of \(ZFC\) in such a way that the <strong>Principle of Dependent Choices</strong> (<a href="/form-classes/howard-rubin-43">Form 43</a>) is true, but no nonprincipal measure exists (<a href="/form-classes/howard-rubin-223">Form 223</a> is false)

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