Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 119, 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: 11016, whose string of implications is:
    191 \(\Rightarrow\) 0
  • A proven non-implication whose code is 5. In this case, it's Code 3: 475, Form 191 \( \not \Rightarrow \) Form 119 whose summary information is:
    Hypothesis Statement
    Form 191 <p> \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). </p>

    Conclusion Statement
    Form 119 <p> <strong>van Douwen's choice principle:</strong> \(C(\aleph_{0}\),uniformly orderable with order type of the integers): Suppose \(\{ A_{i}: i\in\omega\}\) is a set and there is a function \(f\) such that for each \(i\in\omega,\ f(i)\) is an ordering of \(A_{i}\) of type \(\omega^{*}+\omega\) (the usual ordering of the integers), then \(\{A_{i}: i\in\omega\}\) has 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 119 then follows.

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

Name Statement
\(\cal N2(\hbox{LO})\) van Douwen's Model This model is another variationof \(\cal N2\)

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