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This non-implication, Form 131 \( \not \Rightarrow \) Form 357, whose code is 4, 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: 5913, whose string of implications is: 214 \(\Rightarrow\) 76 \(\Rightarrow\) 131
• A proven non-implication whose code is 3. In this case, it's Code 3: 1398, Form 214 \( \not \Rightarrow \) Form 357 whose summary information is:
  

  Hypothesis	Statement
  Form 214	<p> \(Z(\omega)\): For every family \(A\) of infinite sets, there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a non-empty subset of \(y\) and \(|f(y)|=\aleph_{0}\). </p>

  
  

  Conclusion	Statement
  Form 357	<p> \(KW(\aleph_0,\aleph_0)\), <strong>The Kinna-Wagner Selection Principle</strong> for a denumerable family of denumerable sets: For every denumerable set \(M\) of denumerable sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). </p>

  
  
• This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 131 \( \not \Rightarrow \) Form 357 then follows.

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

Name	Statement
\(\cal M29\) Pincus' Model II	Pincus constructs a generic extension \(M[I]\) of a model \(M\) of \(ZF +\) class choice \(+ GCH\) in which \(I=\bigcup_{n\in\omega}I_n\), \(I_{-1}=2\) and \(I_{n+1}\) is a denumerable set of independent functions from \(\omega\) onto \(I_n\)