Hypothesis: HR 121:
\(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function.
Conclusion: HR 323:
\(KW(\infty,WO)\), The Kinna-Wagner Selection Principle for a family of well orderable sets: For every set \(M\) of well orderable sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M29\) Pincus' Model II | Pincus constructs a generic extension \(M[I]\) of a model \(M\) of \(ZF +\) class choice \(+ GCH\) in which \(I=\bigcup_{n\in\omega}I_n\), \(I_{-1}=2\) and \(I_{n+1}\) is a denumerable set of independent functions from \(\omega\) onto \(I_n\) |
| \(\cal N12(\aleph_1)\) A variation of Fraenkel's model, \(\cal N1\) | Thecardinality of \(A\) is \(\aleph_1\), \(\cal G\) is the group of allpermutations on \(A\), and \(S\) is the set of all countable subsets of \(A\).In \(\cal N12(\aleph_1)\), every Dedekind finite set is finite (9 is true),but the \(2m=m\) principle (3) is false |
| \(\cal N24\) 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\(A\) |
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