Implications

This non-implication, Form 90 \( \not \Rightarrow \) Form 430-p, 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: 9778, whose string of implications is:
    89 \(\Rightarrow\) 90
  • A proven non-implication whose code is 5. In this case, it's Code 3: 187, Form 89 \( \not \Rightarrow \) Form 390 whose summary information is:
    Hypothesis Statement
    Form 89 <p> <strong>Antichain Principle:</strong>  Every partially ordered set has a maximal antichain. <a href="/books/8">Jech [1973b]</a>, p 133. </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: 6275, whose string of implications is:
    430-p \(\Rightarrow\) 67 \(\Rightarrow\) 126 \(\Rightarrow\) 82 \(\Rightarrow\) 83 \(\Rightarrow\) 64 \(\Rightarrow\) 390

The conclusion Form 90 \( \not \Rightarrow \) Form 430-p 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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