Implications

This non-implication, Form 133 \( \not \Rightarrow \) Form 15, whose code is 6, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 5. In this case, it's Code 3: 338, Form 133 \( \not \Rightarrow \) Form 45-n whose summary information is:
    Hypothesis Statement
    Form 133  <p> Every set is either well orderable or has an infinite amorphous subset. </p>

    Conclusion 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4271, whose string of implications is:
    15 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 45-n

The conclusion Form 133 \( \not \Rightarrow \) Form 15 then follows.

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

Name Statement
\(\cal N1\) The Basic Fraenkel Model The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)
\(\cal N24\) Hickman's Model I This model is a variation of \(\cal N2\)
\(\cal N24(n)\) An extension of \(\cal N24\) to \(n\)-element sets, \(n>1\).\(A=\bigcup B\), where \( B=\{b_i: i\in\omega\}\) is a pairwise disjoint setof \(n\)-element sets \(\cal G\) is the group of all permutations of \(A\)which are permutations of \(B\); and \(S\) is the set of all finite subsets of\(A\)
\(\cal N26\) Brunner/Pincus Model, a variation of \(\cal N2\) The set ofatoms \(A=\bigcup_{n\in\omega} P_n\), where the \(P_n\)'s are pairwisedisjoint denumerable sets; \(\cal G\) is the set of all permutations\(\sigma\) on \(A\) such that \(\sigma(P_n)=P_n\), for all \(n\in\omega\); and \(S\)is the set of all finite subsets of \(A\)

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