Implications

• Implications Home
• Complete List
• Search

This non-implication, Form 18 \( \not \Rightarrow \) Form 291, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

• An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1559, whose string of implications is: 317 \(\Rightarrow\) 14 \(\Rightarrow\) 153 \(\Rightarrow\) 10 \(\Rightarrow\) 80 \(\Rightarrow\) 18
• A proven non-implication whose code is 5. In this case, it's Code 3: 635, Form 317 \( \not \Rightarrow \) Form 291 whose summary information is:
  

  Hypothesis	Statement
  Form 317	<p> <strong>Weak Sikorski Theorem:</strong>  If \(B\) is a complete, well orderable Boolean algebra and \(f\) is a homomorphism of the Boolean algebra \(A'\) into \(B\) where \(A'\) is a subalgebra of the Boolean algebra \(A\), then \(f\) can be extended to a homomorphism of \(A\) into \(B\).

  
  

  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 18 \( \not \Rightarrow \) 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 \(\le_n\)