Implications

This non-implication, Form 382 \( \not \Rightarrow \) Form 239, whose code is 6, 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: 10026, whose string of implications is:
    381 \(\Rightarrow\) 382
  • A proven non-implication whose code is 5. In this case, it's Code 3: 748, Form 381 \( \not \Rightarrow \) Form 84 whose summary information is:
    Hypothesis Statement
    Form 381 <p> <strong>DUM</strong>:  The disjoint union of metrizable spaces is metrizable. </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: 7928, whose string of implications is:
    239 \(\Rightarrow\) 427 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 77 \(\Rightarrow\) 185 \(\Rightarrow\) 84

The conclusion Form 382 \( \not \Rightarrow \) Form 239 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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