Implications

This non-implication, Form 327 \( \not \Rightarrow \) Form 61, whose code is 4, 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: 9970, whose string of implications is:
    122 \(\Rightarrow\) 327
  • A proven non-implication whose code is 3. In this case, it's Code 3: 246, Form 122 \( \not \Rightarrow \) Form 33-n whose summary information is:
    Hypothesis Statement
    Form 122 <p> \(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function. </p>

    Conclusion Statement
    Form 33-n <p> If \(n\in\omega-\{0,1\}\), \(C(LO,n)\):  Every linearly ordered set of \(n\) element sets has  a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 3960, whose string of implications is:
    61 \(\Rightarrow\) 45-n \(\Rightarrow\) 33-n

The conclusion Form 327 \( \not \Rightarrow \) Form 61 then follows.

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

Name Statement
\(\cal N24(n,LO)\) Truss' Model III This is a variation of \(\cal N24(n)\)in which the set \(B\) is linearly ordered

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