Implications

This non-implication, Form 316 \( \not \Rightarrow \) Form 149, 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: 9704, whose string of implications is:
    51 \(\Rightarrow\) 316
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1052, Form 51 \( \not \Rightarrow \) Form 144 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 144 <p> Every set is almost well orderable. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5153, whose string of implications is:
    149 \(\Rightarrow\) 67 \(\Rightarrow\) 144

The conclusion Form 316 \( \not \Rightarrow \) Form 149 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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