Implications

This non-implication, Form 140 \( \not \Rightarrow \) Form 15, 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: 10205, whose string of implications is:
    88 \(\Rightarrow\) 140
  • A proven non-implication whose code is 3. In this case, it's Code 3: 189, Form 88 \( \not \Rightarrow \) Form 285 whose summary information is:
    Hypothesis Statement
    Form 88 <p>  \(C(\infty ,2)\):  Every family of pairs 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: 1677, whose string of implications is:
    15 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 285

The conclusion Form 140 \( \not \Rightarrow \) Form 15 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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