Implications

This non-implication, Form 80 \( \not \Rightarrow \) Form 50, 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: 1003, whose string of implications is:
    214 \(\Rightarrow\) 9 \(\Rightarrow\) 10 \(\Rightarrow\) 80
  • A proven non-implication whose code is 3. In this case, it's Code 3: 847, Form 214 \( \not \Rightarrow \) Form 47-n 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 47-n <p> If \(n\in\omega-\{0,1\}\), \(C(WO,n)\): Every well ordered collection of \(n\)-element sets has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 2232, whose string of implications is:
    50 \(\Rightarrow\) 14 \(\Rightarrow\) 49 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 121 \(\Rightarrow\) 122 \(\Rightarrow\) 47-n

The conclusion Form 80 \( \not \Rightarrow \) Form 50 then follows.

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

Name Statement
\(\cal M47(n,M)\) Pincus' Model IX This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((E)\)

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