Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 201, 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: 148, Form 0 \( \not \Rightarrow \) Form 80 whose summary information is:
    Hypothesis Statement
    Form 0  \(0 = 0\).

    Conclusion Statement
    Form 80 <p> \(C(\aleph_{0},2)\):  Every denumerable set of  pairs has  a  choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6359, whose string of implications is:
    201 \(\Rightarrow\) 88 \(\Rightarrow\) 80

The conclusion Form 0 \( \not \Rightarrow \) 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
\(\cal N2\) The Second Fraenkel Model The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \).
\(\cal N43\) Brunner's Model II The set of atoms \(A=\bigcup\{P_n: n\in\omega\}\), where \(|P_n|=n+1\) for each \(n\in\omega\) and the \(P_n\)'s arepairwise disjoint

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