Implications

This non-implication, Form 185 \( \not \Rightarrow \) Form 293, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5914, whose string of implications is:
    316 \(\Rightarrow\) 77 \(\Rightarrow\) 185
  • A proven non-implication whose code is 5. In this case, it's Code 3: 614, Form 316 \( \not \Rightarrow \) Form 293 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 293 <p> For all sets \(x\) and \(y\), if \(x\) can be linearly ordered and there is a mapping of \(x\) onto \(y\), then \(y\) can be linearly ordered. </p>

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

The conclusion Form 185 \( \not \Rightarrow \) Form 293 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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