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 \(\Rightarrow\) 49 |
A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar. |
| 49 \(\Rightarrow\) 30 | clear |
| 30 \(\Rightarrow\) 62 | clear |
| 62 \(\Rightarrow\) 61 | clear |
| 61 \(\Rightarrow\) 45-n | clear |
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. |
| 49: | Order Extension Principle: Every partial ordering can be extended to a linear ordering. Tarski [1924], p 78. |
| 30: | Ordering Principle: Every set can be linearly ordered. |
| 62: | \(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function. |
| 61: | \((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element sets has a choice function. |
| 45-n: | If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function. |
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