Implications

This non-implication, Form 12 \( \not \Rightarrow \) Form 28-p, 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: 1253, whose string of implications is:
    3 \(\Rightarrow\) 9 \(\Rightarrow\) 376 \(\Rightarrow\) 377 \(\Rightarrow\) 378 \(\Rightarrow\) 11 \(\Rightarrow\) 12
  • 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: 5787, whose string of implications is:
    28-p \(\Rightarrow\) 427 \(\Rightarrow\) 67 \(\Rightarrow\) 328 \(\Rightarrow\) 330

The conclusion Form 12 \( \not \Rightarrow \) Form 28-p 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 \(ZF + V = L\)

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