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 \(\Rightarrow\) 285 |
On functions without fixed points, Wi'sniewski, K. 1973, Comment. Math. Prace Mat. |
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. |
285: | Let \(E\) be a set and \(f: E\to E\), then \(f\) has a fixed point if and only if \(E\) is not the union of three mutually disjoint sets \(E_1\), \(E_2\) and \(E_3\) such that \(E_i \cap f(E_i) = \emptyset\) for \(i=1, 2, 3\). |
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