Implications

This non-implication, Form 64 \( \not \Rightarrow \) Form 295, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • 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: 2013, whose string of implications is:
    295 \(\Rightarrow\) 30 \(\Rightarrow\) 83

The conclusion Form 64 \( \not \Rightarrow \) Form 295 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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