Implications

This non-implication, Form 337 \( \not \Rightarrow \) Form 89, 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: 4159, whose string of implications is:
    90 \(\Rightarrow\) 51 \(\Rightarrow\) 337
  • A proven non-implication whose code is 5. In this case, it's Code 3: 188, Form 90 \( \not \Rightarrow \) Form 89 whose summary information is:
    Hypothesis Statement
    Form 90 <p> \(LW\):  Every linearly ordered set can be well ordered. <a href="/books/8">Jech [1973b]</a>, p 133. </p>

    Conclusion 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>

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

The conclusion Form 337 \( \not \Rightarrow \) Form 89 then follows.

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

Name Statement
\(\cal N4\) The Mathias/Pincus Model I \(A\) is countably infinite;\(\precsim\) is a universal homogeneous partial ordering on \(A\) (See<a href="/articles/Jech-1973b">Jech [1973b]</a> p 101 for definitions.); \(\cal G\) is the group ofall order automorphisms on \((A,\precsim)\); and \(S\) is the set of allfinite subsets of \(A\)

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