We have the following indirect implication of form equivalence classes:
Implication | Reference |
---|---|
333 \(\Rightarrow\) 67 | clear |
67 \(\Rightarrow\) 114 |
Products of compact spaces in the least permutation model, Brunner, N. 1985a, Z. Math. Logik Grundlagen Math. |
Here are the links and statements of the form equivalence classes referenced above:
Howard-Rubin Number | Statement |
---|---|
333: | \(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is odd. |
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). |
114: | Every A-bounded \(T_2\) topological space is weakly Loeb. (\(A\)-bounded means amorphous subsets are relatively compact. Weakly Loeb means the set of non-empty closed subsets has a multiple choice function.) |
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