Implications

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This non-implication, Form 423 \( \not \Rightarrow \) Form 66, 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: 1165, whose string of implications is: 30 \(\Rightarrow\) 10 \(\Rightarrow\) 423
• A proven non-implication whose code is 3. In this case, it's Code 3: 236, Form 30 \( \not \Rightarrow \) Form 5 whose summary information is:
  

  Hypothesis	Statement
  Form 30	<p> <strong>Ordering Principle:</strong> Every set can be linearly ordered. </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: 5014, whose string of implications is: 66 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 337 \(\Rightarrow\) 92 \(\Rightarrow\) 94 \(\Rightarrow\) 5

The conclusion Form 423 \( \not \Rightarrow \) Form 66 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 \(ZF + V = L\)