Implications

This non-implication, Form 325 \( \not \Rightarrow \) Form 4, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 5. In this case, it's Code 3: 654, Form 325 \( \not \Rightarrow \) Form 84 whose summary information is:
    Hypothesis Statement
    Form 325 <p> <strong>Ramsey's Theorem II:</strong> \(\forall n,m\in\omega\), if A is an infinite set and the family of all \(m\) element subsets of \(A\) is partitioned into \(n\) sets \(S_{j}, 1\le j\le n\), then there is an infinite subset \(B\subseteq A\) such that all \(m\) element subsets of \(B\) belong to the same \(S_{j}\). (Also, see <a href="/form-classes/howard-rubin-17">Form 17</a>.) </p>

    Conclusion Statement
    Form 84 <p> \(E(II,III)\) (<a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): \((\forall x)(x\) is \(T\)-finite  if and only if \(\cal P(x)\) is Dedekind finite). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 96, whose string of implications is:
    4 \(\Rightarrow\) 9 \(\Rightarrow\) 84

The conclusion Form 325 \( \not \Rightarrow \) Form 4 then follows.

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

Name Statement
\(\cal N3\) Mostowski's Linearly Ordered Model \(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all finitesubsets of \(A\)

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