Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 335-n, 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: 138, Form 0 \( \not \Rightarrow \) Form 38 whose summary information is:
    Hypothesis Statement
    Form 0  \(0 = 0\).

    Conclusion Statement
    Form 38 <p> \({\Bbb R}\) is not the union of a countable family of countable sets. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9179, whose string of implications is:
    335-n \(\Rightarrow\) 333 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 337 \(\Rightarrow\) 92 \(\Rightarrow\) 94 \(\Rightarrow\) 5 \(\Rightarrow\) 38

The conclusion Form 0 \( \not \Rightarrow \) Form 335-n then follows.

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

Name Statement
\(\cal M9\) Feferman/Levy Model Assume the ground model, \(\cal M\), satisfies \(ZF + GCH\) (the <strong>Generalized Continuum Hypothesis</strong>)

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