Implications

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This non-implication, Form 104 \( \not \Rightarrow \) Form 106, 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: 10304, whose string of implications is: 34 \(\Rightarrow\) 104
• A proven non-implication whose code is 3. In this case, it's Code 3: 851, Form 34 \( \not \Rightarrow \) Form 5 whose summary information is:
  

  Hypothesis	Statement
  Form 34	<p> \(\aleph_{1}\) is regular. </p>

  
  

  Conclusion	Statement
  Form 5	<p> \(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function. </p>

  
  
• An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7274, whose string of implications is: 106 \(\Rightarrow\) 126 \(\Rightarrow\) 94 \(\Rightarrow\) 5

The conclusion Form 104 \( \not \Rightarrow \) Form 106 then follows.

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

Name	Statement
\(\cal M6\) Sageev's Model I	Using iterated forcing, Sageev constructs \(\cal M6\) by adding a denumerable number of generic tree-like structuresto the ground model, a model of \(ZF + V = L\)