Hypothesis: HR 202:
\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has a choice function.
Conclusion: HR 1:
\(C(\infty,\infty)\): The Axiom of Choice: Every set of non-empty sets has a choice function.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N12(\aleph_1)\) 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 |
Code: 5
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