This non-implication,
Form 0 \( \not \Rightarrow \)
Form 57,
whose code is 4, is constructed around a proven non-implication as follows:
| Hypothesis | Statement |
|---|---|
| Form 0 | \(0 = 0\). |
| 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> |
The conclusion Form 0 \( \not \Rightarrow \) Form 57 then follows.
Finally, the
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M37\) Monro's Model III | This is a generic extension of <a href="/models/Cohen-1">\(\cal M1\)</a> in which there is an amorphous set (<a href="/form-classes/howard-rubin-64">Form 64</a> is false) and \(C(\infty,2)\) (<a href="/form-classes/howard-rubin-88">Form 88</a>) is false |
| \(\cal N1\) 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 \(A\) |
| \(\cal N43\) Brunner's Model II | The set of atoms \(A=\bigcup\{P_n: n\in\omega\}\), where \(|P_n|=n+1\) for each \(n\in\omega\) and the \(P_n\)'s arepairwise disjoint |