Implications

This non-implication, Form 141 \( \not \Rightarrow \) Form 62, 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: 1182, whose string of implications is:
    62 \(\Rightarrow\) 10 \(\Rightarrow\) 358

The conclusion Form 141 \( \not \Rightarrow \) Form 62 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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