We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 407 \(\Rightarrow\) 43 | clear |
| 43 \(\Rightarrow\) 8 | clear |
| 8 \(\Rightarrow\) 16 | clear |
| 16 \(\Rightarrow\) 194 | 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. |
| 8: | \(C(\aleph_{0},\infty)\): |
| 16: | \(C(\aleph_{0},\le 2^{\aleph_{0}})\): Every denumerable collection of non-empty sets each with power \(\le 2^{\aleph_{0}}\) has a choice function. |
| 194: | \(C(\varPi^1_2)\) or \(AC(\varPi^1_2)\): If \(P\in \omega\times{}^{\omega}\omega\), \(P\) has domain \(\omega\), and \(P\) is in \(\varPi^1_2\), then there is a sequence of elements \(\langle x_{k}: k\in\omega\rangle\) of \({}^{\omega}\omega\) with \(\langle k,x_{k}\rangle \in P\) for all \(k\in\omega\). Kanovei [1979]. |
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