Implications

This non-implication, Form 336-n \( \not \Rightarrow \) Form 270, 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: 9983, whose string of implications is:
    45-n \(\Rightarrow\) 336-n
  • A proven non-implication whose code is 3. In this case, it's Code 3: 187, Form 45-n \( \not \Rightarrow \) Form 285 whose summary information is:
    Hypothesis Statement
    Form 45-n <p> If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function. </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: 4746, whose string of implications is:
    270 \(\Rightarrow\) 62 \(\Rightarrow\) 285

The conclusion Form 336-n \( \not \Rightarrow \) Form 270 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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