Implications

This non-implication, Form 165 \( \not \Rightarrow \) Form 264, 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)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7593, whose string of implications is:
    133 \(\Rightarrow\) 231 \(\Rightarrow\) 165
  • A proven non-implication whose code is 5. In this case, it's Code 3: 356, Form 133 \( \not \Rightarrow \) Form 322 whose summary information is:
    Hypothesis Statement
    Form 133  <p> Every set is either well orderable or has an infinite amorphous subset. </p>

    Conclusion Statement
    Form 322 <p> \(KW(WO,\infty)\), <strong>The Kinna-Wagner Selection Principle for a well ordered family of sets:</strong> For every  well ordered set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).  (See <a href="/form-classes/howard-rubin-15">Form 15</a>). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7791, whose string of implications is:
    264 \(\Rightarrow\) 202 \(\Rightarrow\) 40 \(\Rightarrow\) 322

The conclusion Form 165 \( \not \Rightarrow \) Form 264 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 \(A\)

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