Implications

This non-implication, Form 139 \( \not \Rightarrow \) Form 240, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5976, whose string of implications is:
    91 \(\Rightarrow\) 79 \(\Rightarrow\) 139
  • A proven non-implication whose code is 5. In this case, it's Code 3: 222, Form 91 \( \not \Rightarrow \) Form 240 whose summary information is:
    Hypothesis Statement
    Form 91 <p> \(PW\):  The power set of a well ordered set can be well ordered. </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 139 \( \not \Rightarrow \) 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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