Implications

This non-implication, Form 288-n \( \not \Rightarrow \) Form 390, 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: 1730, whose string of implications is:
    325 \(\Rightarrow\) 17 \(\Rightarrow\) 132 \(\Rightarrow\) 10 \(\Rightarrow\) 288-n
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1456, Form 325 \( \not \Rightarrow \) Form 390 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 390 <p> Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements. <a href="/excerpts/Ash-1981-1">Ash [1983]</a>. </p>

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

The conclusion Form 288-n \( \not \Rightarrow \) Form 390 then follows.

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

Name Statement
\(\cal N1\) The Basic Fraenkel Model The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)

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