Implications

This non-implication, Form 306 \( \not \Rightarrow \) Form 353, 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: 6184, whose string of implications is:
    91 \(\Rightarrow\) 79 \(\Rightarrow\) 203 \(\Rightarrow\) 306
  • A proven non-implication whose code is 5. In this case, it's Code 3: 237, Form 91 \( \not \Rightarrow \) Form 350 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 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: 9811, whose string of implications is:
    353 \(\Rightarrow\) 350

The conclusion Form 306 \( \not \Rightarrow \) Form 353 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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