Implications

This non-implication, Form 182 \( \not \Rightarrow \) Form 201, 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: 9618, whose string of implications is:
    191 \(\Rightarrow\) 182
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1446, Form 191 \( \not \Rightarrow \) Form 389 whose summary information is:
    Hypothesis Statement
    Form 191 <p> \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). </p>

    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: 6360, whose string of implications is:
    201 \(\Rightarrow\) 88 \(\Rightarrow\) 80 \(\Rightarrow\) 389

The conclusion Form 182 \( \not \Rightarrow \) Form 201 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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