Implications

This non-implication, Form 190 \( \not \Rightarrow \) Form 303, 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: 7695, whose string of implications is:
    191 \(\Rightarrow\) 189 \(\Rightarrow\) 190
  • A proven non-implication whose code is 5. In this case, it's Code 3: 463, Form 191 \( \not \Rightarrow \) Form 46-K whose summary information is:
    Hypothesis Statement
    Form 191 <p> \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). </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: 4133, whose string of implications is:
    303 \(\Rightarrow\) 50 \(\Rightarrow\) 14 \(\Rightarrow\) 49 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 46-K

The conclusion Form 190 \( \not \Rightarrow \) Form 303 then follows.

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

Name Statement
\(\cal N2(n)\) A generalization of \(\cal N2\) This is a generalization of\(\cal N2\) in which there is a denumerable set of \(n\) element sets for\(n\in\omega - \{0,1\}\)
\(\cal N2(n,M)\) Mostowski's variation of \(\cal N2(n)\) \(A\), \(B\), and \(S\)are the same as in \(\cal N2(n)\)

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