We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 407 \(\Rightarrow\) 43 | clear |
| 43 \(\Rightarrow\) 339 | 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 \(D\cap\ A_i\neq\emptyset\). |
| 43: | \(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136. |
| 339: | Martin's Axiom \((\aleph_{0})\): Whenever \((P\le)\) is a non-empty, ccc quasi-order (ccc means every anti-chain is countable) and \({\cal D}\) is a family of \(\le\aleph_0\) dense subsets of \(P\), then there is a \({\cal D}\) generic filter \(G\) in \(P\). |
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