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 164:
Every non-well-orderable set has an infinite subset with a Dedekind finite power set.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N2\) The Second Fraenkel Model | The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \). |
| \(\cal N2(n)\) A generalization of \(\cal N2\) | This is a generalization of\(\cal N2\) in which there is a denumerable set of \(n\) element sets for\(n\in\omega - \{0,1\}\) |
| \(\cal N2^*(3)\) Howard's variation of \(\cal N2(3)\) | \(A=\bigcup B\), where\(B\) is a set of pairwise disjoint 3 element sets, \(T_i = \{a_i, b_i,c_i\}\) |
| \(\cal N6\) Levy's Model I | \(A=\{a_n : n\in\omega\}\) and \(A = \bigcup \{P_n: n\in\omega\}\), where \(P_0 = \{a_0\}\), \(P_1 = \{a_1,a_2\}\), \(P_2 =\{a_3,a_4,a_5\}\), \(P_3 = \{a_6,a_7,a_8,a_9,a_{10}\}\), \(\cdots\); in generalfor \(n>0\), \(|P_n| = p_n\), where \(p_n\) is the \(n\)th prime |
| \(\cal N50(E)\) 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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