Implications

This non-implication, Form 189 \( \not \Rightarrow \) Form 258, 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: 9615, whose string of implications is:
    191 \(\Rightarrow\) 189
  • 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: 8335, whose string of implications is:
    258 \(\Rightarrow\) 255 \(\Rightarrow\) 260 \(\Rightarrow\) 40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 10 \(\Rightarrow\) 80 \(\Rightarrow\) 389

The conclusion Form 189 \( \not \Rightarrow \) Form 258 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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