Implications

This non-implication, Form 190 \( \not \Rightarrow \) Form 220-p, 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: 494, Form 191 \( \not \Rightarrow \) Form 220-p 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 220-p <p> Suppose \(p\in\omega\) and \(p\) is a prime. Any two elementary Abelian \(p\)-groups (all non-trivial elements have order \(p\)) of the same cardinality are isomorphic. </p>

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

The conclusion Form 190 \( \not \Rightarrow \) Form 220-p then follows.

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

Name Statement
\(\cal N42(p)\) Hickman's Model IV This model is an extension of \(\cal N32\)
\(\cal N45(p)\) Howard/Rubin Model III Let \(p\) be a prime

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