Implications

This non-implication, Form 123 \( \not \Rightarrow \) Form 15, 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: 220, Form 123 \( \not \Rightarrow \) Form 30 whose summary information is:
    Hypothesis Statement
    Form 123 <p> \(SPI^*\): <strong>Uniform weak ultrafilter principle:</strong> For each family \(F\) of infinite sets \(\exists f\) such that \(\forall x\in F\), \(f(x)\) is a non-principal ultrafilter on \(x\). </p>

    Conclusion Statement
    Form 30 <p> <strong>Ordering Principle:</strong> Every set can be linearly ordered. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9791, whose string of implications is:
    15 \(\Rightarrow\) 30

The conclusion Form 123 \( \not \Rightarrow \) Form 15 then follows.

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

Name Statement
\(\cal N7\) L\"auchli's Model I \(A\) is countably infinite

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