We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 407 \(\Rightarrow\) 14 |
Effective equivalents of the Rasiowa-Sikorski lemma, Bacsich, P. D. 1972b, J. London Math. Soc. Ser. 2. |
| 14 \(\Rightarrow\) 99 |
Variants of Rado's selection lemma and their applications, Rav, Y. 1977, Math. Nachr. |
| 99 \(\Rightarrow\) 225 |
Variants of Rado's selection lemma and their applications, Rav, Y. 1977, Math. Nachr. |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 407: | Let \(B\) be a Boolean algebra, \(b\) a non-zero element of \(B\) and \(\{A_i: i\in\omega\}\) a sequence of subsets of \(B\) such that for each \(i\in\omega\), \(A_i\) has a supremum \(a_i\). Then there exists an ultrafilter \(D\) in \(B\) such that \(b\in D\) and, for each \(i\in\omega\), if \(a_i\in D\), then \(D\cap\ A_i\neq\emptyset\). |
| 14: | BPI: Every Boolean algebra has a prime ideal. |
| 99: | Rado's Selection Lemma: Let \(\{K(\lambda): \lambda \in\Lambda\}\) be a family of finite subsets (of \(X\)) and suppose for each finite \(S\subseteq\Lambda\) there is a function \(\gamma(S): S \rightarrow X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))\). Then there is an \(f: \Lambda\rightarrow X\) such that for every finite \(S\subseteq\Lambda\) there is a finite \(T\) such that \(S\subseteq T\subseteq\Lambda\) and such that \(f\) and \(\gamma (T)\) agree on S. |
| 225: | Every proper filter on \(\omega\) can be extended to an ultrafilter. |
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