Implications

This non-implication, Form 199(\(n\)) \( \not \Rightarrow \) Form 238, 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: 6139, whose string of implications is:
    91 \(\Rightarrow\) 79 \(\Rightarrow\) 94 \(\Rightarrow\) 13 \(\Rightarrow\) 199(\(n\))
  • A proven non-implication whose code is 5. In this case, it's Code 3: 221, Form 91 \( \not \Rightarrow \) Form 238 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 238 <p> Every elementary Abelian group (that is, for some prime \(p\) every non identity element has order \(p\)) is the direct sum of cyclic subgroups. </p>

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

The conclusion Form 199(\(n\)) \( \not \Rightarrow \) Form 238 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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