Implications

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This non-implication, Form 425 \( \not \Rightarrow \) Form 264, 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: 5911, whose string of implications is: 214 \(\Rightarrow\) 76 \(\Rightarrow\) 425
• 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: 7761, whose string of implications is: 264 \(\Rightarrow\) 202 \(\Rightarrow\) 40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 32 \(\Rightarrow\) 350

The conclusion Form 425 \( \not \Rightarrow \) Form 264 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 \(I_n\)