Implications

This non-implication, Form 305 \( \not \Rightarrow \) Form 338, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9694, whose string of implications is:
    91 \(\Rightarrow\) 305
  • A proven non-implication whose code is 5. In this case, it's Code 3: 201, Form 91 \( \not \Rightarrow \) Form 119 whose summary information is:
    Hypothesis Statement
    Form 91 <p> \(PW\):  The power set of a well ordered set can be well ordered. </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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 2462, whose string of implications is:
    338 \(\Rightarrow\) 32 \(\Rightarrow\) 119

The conclusion Form 305 \( \not \Rightarrow \) Form 338 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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