Implications

This non-implication, Form 254 \( \not \Rightarrow \) Form 278, 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: 9717, whose string of implications is:
    89 \(\Rightarrow\) 254
  • A proven non-implication whose code is 5. In this case, it's Code 3: 184, Form 89 \( \not \Rightarrow \) Form 278 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 278 <p> In an integral domain \(R\), if every ideal is finitely generated then \(R\) has a maximal proper ideal. <a href="/notes/note-45">note 45</a> E. </p>

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

The conclusion Form 254 \( \not \Rightarrow \) Form 278 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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