In his answer to my question on the real numbers SE/839848, Qiaochu Yuan mentioned that the real numbers are the terminal archimedian ordered field. I wondered, what can be said about the category of archimedian ordered fields. In particular, which limits and colimits do exist? Can the existence of a terminal object be proven categorically?
2026-04-09 18:15:07.1775758507
Categorical proof of the existence of the real numbers
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This category is less interesting than it might sound: because $\mathbf{Q}$ embeds uniquely in its completion $\mathbf{R}$ and is dense in any Archimedean ordered field, there is a unique order-preserving homomorphism from any Archimedean ordered field to $\mathbf{R}$: this category is just the category of subfields of $\mathbf{R}$. And, thus, it is complete and cocomplete, because arbitrary intersections of subfields are again subfields.