Implications

This non-implication, Form 124 \( \not \Rightarrow \) Form 200, 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: 9632, whose string of implications is:
    17 \(\Rightarrow\) 124
  • A proven non-implication whose code is 3. In this case, it's Code 3: 62, Form 17 \( \not \Rightarrow \) Form 200 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 200 <p> For all infinite \(x\), \(|2^{x}| = |x!|\). </p>

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

The conclusion Form 124 \( \not \Rightarrow \) Form 200 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 \(A\)
\(\cal N3\) 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 \(A\)

Edit | Back