We have the following indirect implication of form equivalence classes:
Implication | Reference |
---|---|
123 \(\Rightarrow\) 62 |
Two model theoretic ideas in independence proofs, Pincus, D. 1976, Fund. Math. |
62 \(\Rightarrow\) 10 | clear |
10 \(\Rightarrow\) 80 | clear |
80 \(\Rightarrow\) 18 | clear |
Here are the links and statements of the form equivalence classes referenced above:
Howard-Rubin Number | Statement |
---|---|
123: | \(SPI^*\): Uniform weak ultrafilter principle: For each family \(F\) of infinite sets \(\exists f\) such that \(\forall x\in F\), \(f(x)\) is a non-principal ultrafilter on \(x\). |
62: | \(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function. |
10: | \(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function. |
80: | \(C(\aleph_{0},2)\): Every denumerable set of pairs has a choice function. |
18: | \(PUT(\aleph_{0},2,\aleph_{0})\): The union of a denumerable family of pairwise disjoint pairs has a denumerable subset. |
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