Implications

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This non-implication, Form 327 \( \not \Rightarrow \) Form 27, 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: 7504, whose string of implications is: 165 \(\Rightarrow\) 122 \(\Rightarrow\) 327
• A proven non-implication whose code is 3. In this case, it's Code 3: 876, Form 165 \( \not \Rightarrow \) Form 31 whose summary information is:
  

  Hypothesis	Statement
  Form 165	<p> \(C(WO,WO)\):  Every well ordered family of non-empty, well orderable sets has a choice function. </p>

  
  

  Conclusion	Statement
  Form 31	<p>\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): <strong>The countable union theorem:</strong>  The union of a denumerable set of denumerable sets is denumerable. </p>

  
  
• An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10293, whose string of implications is: 27 \(\Rightarrow\) 31

The conclusion Form 327 \( \not \Rightarrow \) Form 27 then follows.

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

Name	Statement
\(\cal M12(\aleph)\) Truss' Model I	This is a variation of Solovay's model, <a href="/models/Solovay-1">\(\cal M5(\aleph)\)</a> in which \(\aleph\) is singular
\(\cal M20\) Felgner's Model I	Let \(\cal M\) be a model of \(ZF + V = L\). Felgner defines forcing conditions that force \(\aleph_{\omega}\) in \(\cal M\) to be \(\aleph_1\)