Implications

This non-implication, Form 189 \( \not \Rightarrow \) Form 240, 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: 9615, whose string of implications is:
    191 \(\Rightarrow\) 189
  • A proven non-implication whose code is 5. In this case, it's Code 3: 500, Form 191 \( \not \Rightarrow \) Form 240 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 240 <p> If a group \(G\) satisfies "every ascending chain of subgroups is finite", then every subgroup of \(G\) is finitely generated. </p>

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

The conclusion Form 189 \( \not \Rightarrow \) Form 240 then follows.

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

Name Statement
\(\cal N32\) Hickman's Model III This is a variation of \(\cal N1\)

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