Implications

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This non-implication, Form 186 \( \not \Rightarrow \) Form 391, 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: 9770, whose string of implications is: 3 \(\Rightarrow\) 186
• 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: 7444, whose string of implications is: 391 \(\Rightarrow\) 112 \(\Rightarrow\) 395 \(\Rightarrow\) 396 \(\Rightarrow\) 330

The conclusion Form 186 \( \not \Rightarrow \) Form 391 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 \(ZF + V = L\)