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 \(\Rightarrow\) 178-n-N | 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. |
178-n-N: | If \(n\in\omega\), \(n\ge 2\) and \(N \subseteq \{ 1, 2, \ldots , n-1 \}\), \(N \neq\emptyset\), \(MC(\infty,n, N)\): If \(X\) is any set of \(n\)-element sets then there is a function \(f\) with domain \(X\) such that for all \(A\in X\), \(f(A)\subseteq A\) and \(|f(A)|\in N\). |
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