Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 373-n, whose code is 6, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 11308, whose string of implications is:
    337 \(\Rightarrow\) 0
  • A proven non-implication whose code is 5. In this case, it's Code 3: 702, Form 337 \( \not \Rightarrow \) Form 373-n whose summary information is:
    Hypothesis Statement
    Form 337 <p> \(C(WO\), <strong>uniformly linearly ordered</strong>):  If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\). </p>

    Conclusion Statement
    Form 373-n <p> (For \(n\in\omega\), \(n\ge 2\).) \(PC(\aleph_0,n,\infty)\): Every denumerable set of \(n\)-element sets has an infinite subset with a choice function. </p>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 0 \( \not \Rightarrow \) Form 373-n then follows.

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

Name Statement
\(\cal N2(n)\) A generalization of \(\cal N2\) This is a generalization of\(\cal N2\) in which there is a denumerable set of \(n\) element sets for\(n\in\omega - \{0,1\}\)

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