Implications

This non-implication, Form 11 \( \not \Rightarrow \) Form 2, 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: 1108, whose string of implications is:
    214 \(\Rightarrow\) 9 \(\Rightarrow\) 376 \(\Rightarrow\) 377 \(\Rightarrow\) 378 \(\Rightarrow\) 11
  • A proven non-implication whose code is 3. In this case, it's Code 3: 216, Form 214 \( \not \Rightarrow \) Form 3 whose summary information is:
    Hypothesis Statement
    Form 214 <p> \(Z(\omega)\): For every family \(A\) of infinite sets, there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a non-empty subset of \(y\) and \(|f(y)|=\aleph_{0}\). </p>

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

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9882, whose string of implications is:
    2 \(\Rightarrow\) 3

The conclusion Form 11 \( \not \Rightarrow \) Form 2 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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