Implications

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This non-implication, Form 76 \( \not \Rightarrow \) Form 388, 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: 9918, whose string of implications is: 214 \(\Rightarrow\) 76
• A proven non-implication whose code is 3. In this case, it's Code 3: 1382, Form 214 \( \not \Rightarrow \) Form 350 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 350	<p> \(MC(\aleph_0,\aleph_0)\): For every denumerable set \(X\) of non-empty denumerable sets there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). </p>

  
  
• An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7313, whose string of implications is: 388 \(\Rightarrow\) 106 \(\Rightarrow\) 126 \(\Rightarrow\) 350

The conclusion Form 76 \( \not \Rightarrow \) Form 388 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 \(I_n\)