Implications

This non-implication, Form 325 \( \not \Rightarrow \) Form 109, 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: 656, Form 325 \( \not \Rightarrow \) Form 125 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 125 <p> There does not exist an infinite, compact connected \(p\) space. (A \(p\) <em>space</em> is a \(T_2\) space in which the intersection of any well orderable family of open sets is open.) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5045, whose string of implications is:
    109 \(\Rightarrow\) 66 \(\Rightarrow\) 67 \(\Rightarrow\) 144 \(\Rightarrow\) 125

The conclusion Form 325 \( \not \Rightarrow \) Form 109 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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