Implications

This non-implication, Form 219 \( \not \Rightarrow \) Form 165, 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: 2420, whose string of implications is:
    165 \(\Rightarrow\) 32 \(\Rightarrow\) 10

The conclusion Form 219 \( \not \Rightarrow \) Form 165 then follows.

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

Name Statement

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