Implications

This non-implication, Form 390 \( \not \Rightarrow \) Form 4, 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: 10278, whose string of implications is:
    64 \(\Rightarrow\) 390
  • A proven non-implication whose code is 3. In this case, it's Code 3: 205, Form 64 \( \not \Rightarrow \) Form 83 whose summary information is:
    Hypothesis Statement
    Form 64 <p> \(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.) </p>

    Conclusion Statement
    Form 83 <p> \(E(I,II)\) <a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>: \(T\)-finite is equivalent to finite. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 97, whose string of implications is:
    4 \(\Rightarrow\) 9 \(\Rightarrow\) 83

The conclusion Form 390 \( \not \Rightarrow \) Form 4 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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