Hypothesis: HR 67:
\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).
Conclusion: HR 342-n:
(For \(n\in\omega\), \(n\ge 2\).) \(PC(\infty,n,\infty)\): Every infinite family of \(n\)-element sets has an infinite subfamily with a choice function. (See Form 166.)
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N50(E)\) Brunner's Model III | \(E\) is a finite set of prime numbers.For each \(p\in E\) and \(n\in\omega\), let \(A_{p,n}\) be a set of atoms ofcardinality \(p^n\) |
Code: 5
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