Implications

This non-implication, Form 18 \( \not \Rightarrow \) Form 323, 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: 7585, whose string of implications is:
    380 \(\Rightarrow\) 132 \(\Rightarrow\) 10 \(\Rightarrow\) 80 \(\Rightarrow\) 18
  • A proven non-implication whose code is 5. In this case, it's Code 3: 739, Form 380 \( \not \Rightarrow \) Form 64 whose summary information is:
    Hypothesis Statement
    Form 380 <p> \(PC(\infty,WO,\infty)\):  For every infinite family of non-empty well orderable sets, there is an infinite subfamily \(Y\) of \(X\) which has a choice function. </p>

    Conclusion Statement
    Form 64 <p> \(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4532, whose string of implications is:
    323 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 11 \(\Rightarrow\) 12 \(\Rightarrow\) 336-n \(\Rightarrow\) 64

The conclusion Form 18 \( \not \Rightarrow \) Form 323 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 \(A\)
\(\cal N26\) 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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