Implications

This non-implication, Form 76 \( \not \Rightarrow \) Form 344, 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: 9918, whose string of implications is:
    214 \(\Rightarrow\) 76
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1373, Form 214 \( \not \Rightarrow \) Form 344 whose summary information is:
    Hypothesis Statement
    Form 214 <p> \(Z(\omega)\): For every family \(A\) of infinite sets, there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a non-empty subset of \(y\) and \(|f(y)|=\aleph_{0}\). </p>

    Conclusion Statement
    Form 344 <p> If \((E_i)_{i\in I}\) is a family of non-empty sets, then there is a family \((U_i)_{i\in I}\) such that \(\forall i\in I\), \(U_i\) is an ultrafilter on \(E_i\). </p>

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

The conclusion Form 76 \( \not \Rightarrow \) Form 344 then follows.

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

Name Statement
\(\cal M46(m,M)\) Pincus' Model VIII This model depends on the natural number \(m\) and the set of natural numbers \(M\) which must satisfy Mostowski's condition: <ul type="none"> <li>\(S(M,m)\): For everydecomposition \(m = p_{1} + \ldots + p_{s}\) of \(m\) into a sum of primes at least one \(p_{i}\) divides an element of \(M\)</li> </ul>
\(\cal M47(n,M)\) Pincus' Model IX This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((E)\)

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