Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 139, 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: 168, Form 0 \( \not \Rightarrow \) Form 389 whose summary information is:
    Hypothesis Statement
    Form 0  \(0 = 0\).

    Conclusion Statement
    Form 389 <p> \(C(\aleph_0,2,\cal P({\Bbb R}))\): Every denumerable family of two element subsets of \(\cal P({\Bbb R})\) has a choice function.  \ac{Keremedis} \cite{1999b}. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9806, whose string of implications is:
    139 \(\Rightarrow\) 389

The conclusion Form 0 \( \not \Rightarrow \) Form 139 then follows.

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

Name Statement
\(\cal M7\) Cohen's Second Model There are two denumerable subsets\(U=\{U_i:i\in\omega\}\) and \(V=\{V_i:i\in\omega\}\) of \(\cal P({\Bbb R})\)(neither of which is in the model) such that for each \(i\in\omega\), \(U_i\)and \(V_i\) cannot be distinguished in the model

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