Implications

This non-implication, Form 415 \( \not \Rightarrow \) Form 397, 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: 9668, whose string of implications is:
    144 \(\Rightarrow\) 415
  • A proven non-implication whose code is 5. In this case, it's Code 3: 414, Form 144 \( \not \Rightarrow \) Form 350 whose summary information is:
    Hypothesis Statement
    Form 144 <p> Every set is almost well orderable. </p>

    Conclusion Statement
    Form 350 <p> \(MC(\aleph_0,\aleph_0)\): For every denumerable set \(X\) of non-empty denumerable sets there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9073, whose string of implications is:
    397 \(\Rightarrow\) 330 \(\Rightarrow\) 350

The conclusion Form 415 \( \not \Rightarrow \) Form 397 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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