Implications

This non-implication, Form 191 \( \not \Rightarrow \) Form 68, whose code is 6, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 5. In this case, it's Code 3: 517, Form 191 \( \not \Rightarrow \) Form 342-n 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 342-n <p> (For \(n\in\omega\), \(n\ge 2\).) \(PC(\infty,n,\infty)\):  Every infinite family of \(n\)-element sets has an infinite subfamily with a choice function. (See <a href="/form-classes/howard-rubin-166">Form 166</a>.) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4483, whose string of implications is:
    68 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 11 \(\Rightarrow\) 12 \(\Rightarrow\) 73 \(\Rightarrow\) 342-n

The conclusion Form 191 \( \not \Rightarrow \) Form 68 then follows.

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

Name Statement
\(\cal N50(E)\) Brunner's Model III \(E\) is a finite set of prime numbers.For each \(p\in E\) and \(n\in\omega\), let \(A_{p,n}\) be a set of atoms ofcardinality \(p^n\)

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