Implications

This non-implication, Form 389 \( \not \Rightarrow \) Form 215, 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: 1722, whose string of implications is:
    17 \(\Rightarrow\) 132 \(\Rightarrow\) 10 \(\Rightarrow\) 80 \(\Rightarrow\) 389
  • A proven non-implication whose code is 3. In this case, it's Code 3: 912, Form 17 \( \not \Rightarrow \) Form 64 whose summary information is:
    Hypothesis Statement
    Form 17 <p> <strong>Ramsey's Theorem I:</strong> If \(A\) is an infinite set and the family of all 2 element subsets of \(A\) is partitioned into 2 sets \(X\) and \(Y\), then there is an infinite subset \(B\subseteq A\) such that all 2 element subsets of \(B\) belong to \(X\) or all 2 element subsets of \(B\) belong to \(Y\). (Also, see <a href="/form-classes/howard-rubin-325">Form 325</a>.), <a href="/books/8">Jech [1973b]</a>, p 164 prob 11.20. </p>

    Conclusion 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6282, whose string of implications is:
    215 \(\Rightarrow\) 83 \(\Rightarrow\) 64

The conclusion Form 389 \( \not \Rightarrow \) Form 215 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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