Implications

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This non-implication, Form 190 \( \not \Rightarrow \) Form 329, whose code is 6, 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: 7695, whose string of implications is: 191 \(\Rightarrow\) 189 \(\Rightarrow\) 190
• A proven non-implication whose code is 5. In this case, it's Code 3: 519, Form 191 \( \not \Rightarrow \) Form 350 whose summary information is:
  

  Hypothesis	Statement
  Form 191	<p> \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). </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: 9071, whose string of implications is: 329 \(\Rightarrow\) 330 \(\Rightarrow\) 350

The conclusion Form 190 \( \not \Rightarrow \) Form 329 then follows.

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

Name	Statement
\(\cal N41\) Another variation of \(\cal N3\)	\(A=\bigcup\{B_n; n\in\omega\}\)is a disjoint union, where each \(B_n\) is denumerable and ordered like therationals by \(\le_n\)