Implications

This non-implication, Form 405 \( \not \Rightarrow \) Form 343, 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: 7622, whose string of implications is:
    214 \(\Rightarrow\) 152 \(\Rightarrow\) 4 \(\Rightarrow\) 405
  • A proven non-implication whose code is 3. In this case, it's Code 3: 889, Form 214 \( \not \Rightarrow \) Form 46-K 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 46-K <p> If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(\infty,K)\): For every \(n\in K\), every set 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: 4595, whose string of implications is:
    343 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 46-K

The conclusion Form 405 \( \not \Rightarrow \) Form 343 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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