Implications

This non-implication, Form 276 \( \not \Rightarrow \) Form 295, 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: 9765, whose string of implications is:
    88 \(\Rightarrow\) 276
  • 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: 2067, whose string of implications is:
    295 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 285

The conclusion Form 276 \( \not \Rightarrow \) Form 295 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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