Implications

This non-implication, Form 38 \( \not \Rightarrow \) Form 91, 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: 1916, whose string of implications is:
    51 \(\Rightarrow\) 25 \(\Rightarrow\) 34 \(\Rightarrow\) 38
  • 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>

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

The conclusion Form 38 \( \not \Rightarrow \) Form 91 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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