Implications

• Implications Home
• Complete List
• Search

This non-implication, Form 23 \( \not \Rightarrow \) Form 278, 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: 63, Form 23 \( \not \Rightarrow \) Form 278 whose summary information is:
  

  Hypothesis	Statement
  Form 23	<p> \((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\). </p>

  
  

  Conclusion	Statement
  Form 278	<p> In an integral domain \(R\), if every ideal is finitely generated then \(R\) has a maximal proper ideal. <a href="/notes/note-45">note 45</a> E. </p>

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

The conclusion Form 23 \( \not \Rightarrow \) Form 278 then follows.

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

Name	Statement
\(\cal N1\) The Basic Fraenkel Model	The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)