Implications

This non-implication, Form 290 \( \not \Rightarrow \) Form 259, 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: 1080, Form 290 \( \not \Rightarrow \) Form 151 whose summary information is:
    Hypothesis Statement
    Form 290 <p> For all infinite \(x\), \(|2^x|=|x^x|\). </p>

    Conclusion Statement
    Form 151 <p> \(UT(WO,\aleph_{0},WO)\) (\(U_{\aleph_{1}}\)): The union of a well ordered set of denumerable sets is well  orderable. (If \(\kappa\) is a well ordered cardinal, see <a href="/notes/note-27">note 27</a> for \(UT(WO,\kappa,WO)\).) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8880, whose string of implications is:
    259 \(\Rightarrow\) 260 \(\Rightarrow\) 40 \(\Rightarrow\) 231 \(\Rightarrow\) 151

The conclusion Form 290 \( \not \Rightarrow \) Form 259 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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