Implications

This non-implication, Form 358 \( \not \Rightarrow \) Form 294, 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: 1159, whose string of implications is:
    122 \(\Rightarrow\) 10 \(\Rightarrow\) 358
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1251, Form 122 \( \not \Rightarrow \) Form 294 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 294 <p> Every linearly ordered \(W\)-set is well orderable. </p>

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

The conclusion Form 358 \( \not \Rightarrow \) Form 294 then follows.

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

Name Statement
\(\cal N41\) Another variation of \(\cal N3\) \(A=\bigcup\{B_n; n\in\omega\}\)is a disjoint union, where each \(B_n\) is denumerable and ordered like therationals by \(\le_n\)

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