Implications

This non-implication, Form 26 \( \not \Rightarrow \) Form 398, 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: 9620, whose string of implications is:
    24 \(\Rightarrow\) 26
  • A proven non-implication whose code is 5. In this case, it's Code 3: 99, Form 24 \( \not \Rightarrow \) Form 355 whose summary information is:
    Hypothesis Statement
    Form 24 <p> \(C(\aleph_0,2^{(2^{\aleph_0})})\): Every denumerable collection of non-empty sets each with power \(2^{(2^{\aleph_{0}})}\) has a choice function. </p>

    Conclusion Statement
    Form 355 <p> \(KW(\aleph_0,\infty)\), <strong>The Kinna-Wagner Selection Principle</strong> for a denumerable family of sets: For every denumerable set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8937, whose string of implications is:
    398 \(\Rightarrow\) 322 \(\Rightarrow\) 355

The conclusion Form 26 \( \not \Rightarrow \) Form 398 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 \(A\)
\(\cal N26\) Brunner/Pincus Model, a variation of \(\cal N2\) The set ofatoms \(A=\bigcup_{n\in\omega} P_n\), where the \(P_n\)'s are pairwisedisjoint denumerable sets; \(\cal G\) is the set of all permutations\(\sigma\) on \(A\) such that \(\sigma(P_n)=P_n\), for all \(n\in\omega\); and \(S\)is the set of all finite subsets of \(A\)

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