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This non-implication, Form 16 \( \not \Rightarrow \) Form 291, 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: 28, Form 16 \( \not \Rightarrow \) Form 291 whose summary information is:
  

  Hypothesis	Statement
  Form 16	<p> \(C(\aleph_{0},\le 2^{\aleph_{0}})\):  Every denumerable collection of non-empty sets  each with power \(\le  2^{\aleph_{0}}\) has a choice function. </p>

  
  

  Conclusion	Statement
  Form 291	<p> For all infinite \(x\), \(|x!|=|x^x|\). </p>

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

The conclusion Form 16 \( \not \Rightarrow \) Form 291 then follows.

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

Name	Statement
\(\cal N29\) Dawson/Howard Model	Let \(A=\bigcup\{B_n; n\in\omega\}\) is a disjoint union, where each \(B_n\) is denumerable and ordered like the rationals by \(\le_n\)