Implications

This non-implication, Form 206 \( \not \Rightarrow \) Form 337, 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: 5940, whose string of implications is:
    91 \(\Rightarrow\) 79 \(\Rightarrow\) 70 \(\Rightarrow\) 206
  • A proven non-implication whose code is 5. In this case, it's Code 3: 235, Form 91 \( \not \Rightarrow \) Form 337 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 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>

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

The conclusion Form 206 \( \not \Rightarrow \) Form 337 then follows.

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

Name Statement
\(\cal N53\) Good/Tree/Watson Model I Let \(A=\bigcup \{Q_n:\ n\in\omega\}\), where \(Q_n=\{a_{n,q}:q\in \Bbb{Q}\}\)

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