Implications

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This non-implication, Form 405 \( \not \Rightarrow \) Form 40, 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: 49, whose string of implications is: 3 \(\Rightarrow\) 4 \(\Rightarrow\) 405
• A proven non-implication whose code is 3. In this case, it's Code 3: 1334, Form 3 \( \not \Rightarrow \) Form 330 whose summary information is:
  

  Hypothesis	Statement
  Form 3	\(2m = m\): For all infinite cardinals \(m\), \(2m = m\).

  
  

  Conclusion	Statement
  Form 330	<p> \(MC(WO,WO)\): For every well ordered set \(X\) of well orderable sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\).  (See <a href="/form-classes/howard-rubin-67">Form 67</a>.) </p>

  
  
• An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 3470, whose string of implications is: 40 \(\Rightarrow\) 165 \(\Rightarrow\) 330

The conclusion Form 405 \( \not \Rightarrow \) Form 40 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 \(ZF + V = L\)