Implications

This non-implication, Form 309 \( \not \Rightarrow \) Form 408, 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: 9695, whose string of implications is:
    91 \(\Rightarrow\) 309
  • A proven non-implication whose code is 5. In this case, it's Code 3: 227, Form 91 \( \not \Rightarrow \) Form 285 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 285 <p> Let \(E\) be a set and \(f: E\to E\), then \(f\) has a fixed point if and only if \(E\) is not the union of three mutually disjoint sets \(E_1\), \(E_2\) and \(E_3\) such that \(E_i \cap f(E_i) = \emptyset\) for \(i=1, 2, 3\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4782, whose string of implications is:
    408 \(\Rightarrow\) 62 \(\Rightarrow\) 285

The conclusion Form 309 \( \not \Rightarrow \) Form 408 then follows.

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

Name Statement
\(\cal N22(p)\) Makowski/Wi\'sniewski/Mostowski Model (Where \(p\) is aprime) Let \(A=\bigcup\{A_i: i\in\omega\}\) where The \(A_i\)'s are pairwisedisjoint and each has cardinality \(p\)

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