Implications

This non-implication, Form 197 \( \not \Rightarrow \) Form 97, 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: 6811, whose string of implications is:
    164 \(\Rightarrow\) 91 \(\Rightarrow\) 79 \(\Rightarrow\) 197
  • A proven non-implication whose code is 5. In this case, it's Code 3: 434, Form 164 \( \not \Rightarrow \) Form 97 whose summary information is:
    Hypothesis Statement
    Form 164 <p> Every non-well-orderable set has an infinite subset with a Dedekind finite power set. </p>

    Conclusion Statement
    Form 97 <p> <strong>Cardinal Representatives:</strong> For every set \(A\) there is a function \(c\) with domain \({\cal P}(A)\) such that for all \(x, y\in {\cal P}(A)\), (i) \(c(x) = c(y) \leftrightarrow x\approx y\) and (ii) \(c(x)\approx x\).  <a href="/books/8">Jech [1973b]</a> p 154. </p>

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

The conclusion Form 197 \( \not \Rightarrow \) Form 97 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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