Implications

This non-implication, Form 77 \( \not \Rightarrow \) Form 51, whose code is 6, 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: 9700, whose string of implications is:
    316 \(\Rightarrow\) 77
  • A proven non-implication whose code is 5. In this case, it's Code 3: 611, Form 316 \( \not \Rightarrow \) Form 51 whose summary information is:
    Hypothesis Statement
    Form 316 <p> If a linearly ordered set \((A,\le)\) has the fixed point property then \((A,\le)\) is complete. (\((A,\le)\)  has the <em>fixed point property</em> if every function \(f:A\to A\) satisfying \((x\le y \Rightarrow f(x)\le f(y))\) has a fixed point, and (\((A,\le)\) is <em>complete</em> if every subset of \(A\) has a least upper bound.) </p>

    Conclusion 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>

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

The conclusion Form 77 \( \not \Rightarrow \) Form 51 then follows.

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

Name Statement
\(\cal N37\) A variation of Blass' model, \(\cal N28\) Let \(A=\{a_{i,j}:i\in\omega, j\in\Bbb Z\}\)

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