Implications

This non-implication, Form 315 \( \not \Rightarrow \) Form 391, whose code is 4, 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: 1911, whose string of implications is:
    51 \(\Rightarrow\) 25 \(\Rightarrow\) 315
  • A proven non-implication whose code is 3. In this case, it's Code 3: 937, Form 51 \( \not \Rightarrow \) Form 91 whose summary information is:
    Hypothesis Statement
    Form 51 <p> <strong>Cofinality Principle:</strong> Every linear ordering has a cofinal sub well ordering.  <a href="/articles/Sierpi\'nski-1918">Sierpi\'nski [1918]</a>, p 117. </p>

    Conclusion Statement
    Form 91 <p> \(PW\):  The power set of a well ordered set can be well ordered. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7424, whose string of implications is:
    391 \(\Rightarrow\) 112 \(\Rightarrow\) 90 \(\Rightarrow\) 91

The conclusion Form 315 \( \not \Rightarrow \) Form 391 then follows.

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

Name Statement
\(\cal M14\) Morris' Model I This is an extension of Mathias' model, <a href="/models/Mathias-1">\(\cal M3\)</a>

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