# Categorical proof of the existence of the real numbers

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?

• What is a "categorical proof"? There are a number of general principles that tell you that some category is complete if another is, but at some point one actually has to sit down and prove that a particular category is complete. I think this is one of those cases. – Zhen Lin Jun 19 '14 at 19:30

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.
• Of course you have said this, but just to be clear: This category is (equivalent to) a partial order. This follows simply because $\mathbb{Q}$ is dense in any Archimedean ordered field, so that any two parallel homomorphisms agree. We don't need $\mathbb{R}$ for this reasoning. – Martin Brandenburg Jun 19 '14 at 23:28