Implications

This non-implication, Form 290 \( \not \Rightarrow \) Form 86-alpha, 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: 227, Form 290 \( \not \Rightarrow \) Form 32 whose summary information is:
    Hypothesis Statement
    Form 290 <p> For all infinite \(x\), \(|2^x|=|x^x|\). </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: 694, whose string of implications is:
    86-alpha \(\Rightarrow\) 8 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 32

The conclusion Form 290 \( \not \Rightarrow \) Form 86-alpha 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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