Implications

This non-implication, Form 216 \( \not \Rightarrow \) Form 9, whose code is 4, 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 3. In this case, it's Code 3: 207, Form 216 \( \not \Rightarrow \) Form 217 whose summary information is:
    Hypothesis Statement
    Form 216 <p> Every infinite tree has either an infinite chain or an infinite antichain. </p>

    Conclusion Statement
    Form 217 <p> Every infinite partially ordered set has either an infinite chain or an infinite antichain. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10225, whose string of implications is:
    9 \(\Rightarrow\) 217

The conclusion Form 216 \( \not \Rightarrow \) Form 9 then follows.

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

Name Statement
\(\cal N34\) Pincus' Model X For each \(q\in\Bbb Q\), let \(C_q=\{a_q,b_q\}\), a pair of atoms and let \(A=\bigcup_{q\in\Bbb Q}C_q\)

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