Implications

This non-implication, Form 164 \( \not \Rightarrow \) Form 174-alpha, whose code is 6, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 5. In this case, it's Code 3: 438, Form 164 \( \not \Rightarrow \) Form 131 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 131 <p> \(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 3670, whose string of implications is:
    174-alpha \(\Rightarrow\) 43 \(\Rightarrow\) 8 \(\Rightarrow\) 126 \(\Rightarrow\) 131

The conclusion Form 164 \( \not \Rightarrow \) Form 174-alpha 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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