Implications

This non-implication, Form 219 \( \not \Rightarrow \) Form 132, 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: 202, Form 219 \( \not \Rightarrow \) Form 10 whose summary information is:
    Hypothesis Statement
    Form 219 <p> \((\forall n\in\omega-\{0\}) MC(\infty,WO\), relatively prime to \(n\)): For all non-zero \(n\in \omega\), if \(X\) is a set of non-empty well orderable sets, then there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a non-empty, finite subset of \(x\), and \(|f(x)|\) is relatively prime to \(n\). </p>

    Conclusion Statement
    Form 10 <p> \(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9634, whose string of implications is:
    132 \(\Rightarrow\) 10

The conclusion Form 219 \( \not \Rightarrow \) Form 132 then follows.

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

Name Statement

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