Implications

This non-implication, Form 141 \( \not \Rightarrow \) Form 165, 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: 406, Form 141 \( \not \Rightarrow \) Form 358 whose summary information is:
    Hypothesis Statement
    Form 141 <p> <a href="/form-class-members/howard-rubin-14-p-n">[14 P(\(n\))]</a> with \(n = 2\):  Let \(\{A(i): i\in I\}\) be a collection of sets such that \(\forall i\in I,\ |A(i)|\le 2\) and suppose \(R\) is a symmetric binary relation on \(\bigcup^{}_{i\in I} A(i)\) such that for all finite \(W\subseteq I\) there is an \(R\) consistent choice function for \(\{A(i): i \in W\}\). Then there is an \(R\) consistent choice function for \(\{A(i): i\in I\}\). </p>

    Conclusion Statement
    Form 358 <p> \(KW(\aleph_0,<\aleph_0)\), <strong>The Kinna-Wagner Selection Principle</strong> for a denumerable family of finite sets: For every denumerable set \(M\) of finite sets 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: 2421, whose string of implications is:
    165 \(\Rightarrow\) 32 \(\Rightarrow\) 10 \(\Rightarrow\) 358

The conclusion Form 141 \( \not \Rightarrow \) Form 165 then follows.

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

Name Statement
\(\cal N2^*(3)\) Howard's variation of \(\cal N2(3)\) \(A=\bigcup B\), where\(B\) is a set of pairwise disjoint 3 element sets, \(T_i = \{a_i, b_i,c_i\}\)

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