Implications

This non-implication, Form 200 \( \not \Rightarrow \) Form 421, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 218, Form 200 \( \not \Rightarrow \) Form 32 whose summary information is:
    Hypothesis Statement
    Form 200 <p> For all infinite \(x\), \(|2^{x}| = |x!|\). </p>

    Conclusion Statement
    Form 32 <p> \(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets  has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9345, whose string of implications is:
    421 \(\Rightarrow\) 338 \(\Rightarrow\) 32

The conclusion Form 200 \( \not \Rightarrow \) Form 421 then follows.

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

Name Statement
\(\cal M29\) Pincus' Model II Pincus constructs a generic extension \(M[I]\) of a model \(M\) of \(ZF +\) class choice \(+ GCH\) in which \(I=\bigcup_{n\in\omega}I_n\), \(I_{-1}=2\) and \(I_{n+1}\) is a denumerable set of independent functions from \(\omega\) onto \(I_n\)

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