David Corfield The Reals

• John Armstrong

It turns out that the real numbers are actually the largest Archimedean field. That is, if $\mathbb{F}$ is any ordered field satisfying the Archimedean property, there will be an monomorphism of ordered fields $\mathbb{F}\rightarrow\mathbb{R}$, making (the image of) $\mathbb{F}$ a subfield of $\mathbb{R}$. I won’t prove this here, but I will note one thing about the meaning of this result: the Archimedean property essentially limits the size of an ordered field. That is, an ordered field can’t get too big without breaking this property. Dually, an ordered field can’t get too small without breaking Dedekind completeness or uniform completeness. Completeness pulls the field one way, while the Archimedean property pulls the other way, and the two reach a sort of equilibrium in the real numbers, living both at the top of one world and the bottom of the other.

• The completion of $\mathbb{Q}$ as a uniform space is the (underlying uniform space of) $\mathbb{R}$. Completion is left adjoint to the inclusion of $Unif$ in $CUnif$.

• The complete ordered field.

• Terminal object in the category of Archimedean fields.

• Here: irreducible locally compact abelian group with no compace subgroup.

• The reals have an algebraic completion of finite degree, which is unique and metrically complete.