Implications

This non-implication, Form 309 \( \not \Rightarrow \) Form 295, 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: 231, Form 91 \( \not \Rightarrow \) Form 293 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 293 <p> For all sets \(x\) and \(y\), if \(x\) can be linearly ordered and there is a mapping of \(x\) onto \(y\), then \(y\) can be linearly ordered. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 2010, whose string of implications is:
    295 \(\Rightarrow\) 30 \(\Rightarrow\) 293

The conclusion Form 309 \( \not \Rightarrow \) Form 295 then follows.

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

Name Statement
\(\cal N28\) Blass' Permutation Model The set \(A=\{a^i_{\xi}: i\in \Bbb Z, \xi\in\aleph_1\}\)
\(\cal N37\) A variation of Blass' model, \(\cal N28\) Let \(A=\{a_{i,j}:i\in\omega, j\in\Bbb Z\}\)

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