We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 212 \(\Rightarrow\) 369 |
Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart. |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 212: | \(C(2^{\aleph_{0}},\subseteq{\Bbb R})\): If \(R\) is a relation on \({\Bbb R}\) such that for all \(x\in{\Bbb R}\), there is a \(y\in{\Bbb R}\) such that \(x\mathrel R y\), then there is a function \(f: {\Bbb R} \rightarrow{\Bbb R}\) such that for all \(x\in{\Bbb R}\), \(x\mathrel R f(x)\). |
| 369: | If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\). |
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