Implications

This non-implication, Form 165 \( \not \Rightarrow \) Form 129, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • 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: 449, Form 165 \( \not \Rightarrow \) Form 167 whose summary information is:
    Hypothesis Statement
    Form 165 <p> \(C(WO,WO)\):  Every well ordered family of non-empty, well orderable sets has a choice function. </p>

    Conclusion Statement
    Form 167 <p> \(PKW(\aleph_{0},\ge 2,\infty)\), <strong>Partial Kinna-Wagner Principle:</strong>  For every denumerable family \(F\) such that for all \(x\in F\), \(|x|\ge 2\), there is an infinite subset \(H\subseteq F\) and a function \(f\) such that for all \(x\in H\), \(\emptyset\neq f(x) \subsetneq x\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 208, whose string of implications is:
    129 \(\Rightarrow\) 4 \(\Rightarrow\) 9 \(\Rightarrow\) 376 \(\Rightarrow\) 167

The conclusion Form 165 \( \not \Rightarrow \) Form 129 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal N26\) 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 \(A\)
\(\cal N29\) Dawson/Howard Model Let \(A=\bigcup\{B_n; n\in\omega\}\) is a disjoint union, where each \(B_n\) is denumerable and ordered like the rationals by \(\le_n\)

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