Implications

This non-implication, Form 309 \( \not \Rightarrow \) Form 392, 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: 9695, whose string of implications is:
    91 \(\Rightarrow\) 309
  • 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9518, whose string of implications is:
    392 \(\Rightarrow\) 394 \(\Rightarrow\) 337

The conclusion Form 309 \( \not \Rightarrow \) Form 392 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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