Implications

This non-implication, Form 316 \( \not \Rightarrow \) Form 112, whose code is 6, is constructed around a proven non-implication as follows:

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

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6514, whose string of implications is:
    112 \(\Rightarrow\) 90 \(\Rightarrow\) 51

The conclusion Form 316 \( \not \Rightarrow \) Form 112 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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