Implications

This non-implication, Form 216 \( \not \Rightarrow \) Form 405, 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: 10311, whose string of implications is:
    217 \(\Rightarrow\) 216
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1452, Form 217 \( \not \Rightarrow \) Form 390 whose summary information is:
    Hypothesis Statement
    Form 217 <p> Every infinite partially ordered set has either an infinite chain or an infinite antichain. </p>

    Conclusion Statement
    Form 390 <p> Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements. <a href="/excerpts/Ash-1981-1">Ash [1983]</a>. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5908, whose string of implications is:
    405 \(\Rightarrow\) 75 \(\Rightarrow\) 404 \(\Rightarrow\) 390

The conclusion Form 216 \( \not \Rightarrow \) Form 405 then follows.

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

Name Statement
\(\cal N1\) The Basic Fraenkel Model The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)

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