Implications

This non-implication, Form 253 \( \not \Rightarrow \) Form 225, 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: 5889, whose string of implications is:
    225 \(\Rightarrow\) 70 \(\Rightarrow\) 206

The conclusion Form 253 \( \not \Rightarrow \) Form 225 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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