Implications

This non-implication, Form 173 \( \not \Rightarrow \) Form 76, whose code is 6, 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: 9747, whose string of implications is:
    232 \(\Rightarrow\) 173
  • A proven non-implication whose code is 5. In this case, it's Code 3: 551, Form 232 \( \not \Rightarrow \) Form 76 whose summary information is:
    Hypothesis Statement
    Form 232 <p> Every metric space \((X,d)\) has a \(\sigma\)-point finite base. </p>

    Conclusion Statement
    Form 76 <p> \(MC_\omega(\infty,\infty)\) (\(\omega\)-MC): For every family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\). </p>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 173 \( \not \Rightarrow \) Form 76 then follows.

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

Name Statement
\(\cal N1\) The Basic Fraenkel Model The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)

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