Uniqueness of the complete ordered field: $\mathbb{R}$ up to a unique isomorphism

For each $x\in F$, applying [L3] (with $F$ Archimedean by [L1]) to $x-1<x$ and to $x<x+1$ gives rationals $a,b$ with ${\iota }_F(a)<x<{\iota }_F(b)$; then ${\iota }_G(a)\in L_x$, and every ${\iota }_G(q)\in L_x$ has ${\iota }_F(q)<{\iota }_F(b)$ so $q<b$ hence ${\iota }_G(q)<{\iota }_G(b)$, so $L_x$ is nonempty and bounded above and $\phi (x)=\sup L_x$ exists in $G$ by [L6].

If $x>0$, then applying [L3] to $0<x$ gives a rational $q$ with $0<{\iota }_F(q)<x$, whence $q>0$, ${\iota }_G(q)>0$, and ${\iota }_G(q)\in L_x$, so $\phi (x)=\sup L_x\ge {\iota }_G(q)>0$.

For $r\in \mathbb{Q}$, $L_{{\iota }_F(r)}=\{{\iota }_G(q):q<r\}$ by [L2]; ${\iota }_G(r)$ is an upper bound, and any $w<{\iota }_G(r)$ is exceeded by some ${\iota }_G(q^{\mathrm{\prime }})\in L_{{\iota }_F(r)}$ via density [L3] in $G$, so $\sup ={\iota }_G(r)$, i.e. $\phi ({\iota }_F(r))={\iota }_G(r)$.

For rationals $q,q^{\mathrm{\prime }}$ with ${\iota }_F(q)<x$ and ${\iota }_F(q^{\mathrm{\prime }})<x^{\mathrm{\prime }}$, additivity of ${\iota }_F$ gives ${\iota }_F(q+q^{\mathrm{\prime }})<x+x^{\mathrm{\prime }}$, so ${\iota }_G(q)+{\iota }_G(q^{\mathrm{\prime }})={\iota }_G(q+q^{\mathrm{\prime }})\in L_{x+x^{\mathrm{\prime }}}$ and $\phi (x+x^{\mathrm{\prime }})\ge {\iota }_G(q)+{\iota }_G(q^{\mathrm{\prime }})$; fixing $q^{\mathrm{\prime }}$ and taking the sup over $q$, then over $q^{\mathrm{\prime }}$, yields $\phi (x+x^{\mathrm{\prime }})\ge \phi (x)+\phi (x^{\mathrm{\prime }})$ by the least-upper-bound calculus [L8].

For any rational $s$ with ${\iota }_F(s)<x+x^{\mathrm{\prime }}$ we have ${\iota }_F(s)-x^{\mathrm{\prime }}<x$; density [L3] gives a rational $q$ with ${\iota }_F(s)-x^{\mathrm{\prime }}<{\iota }_F(q)<x$, so ${\iota }_F(q)<x$, and the left inequality gives ${\iota }_F(s)-{\iota }_F(q)<x^{\mathrm{\prime }}$, i.e. ${\iota }_F(s-q)={\iota }_F(s)-{\iota }_F(q)<x^{\mathrm{\prime }}$ by additivity of ${\iota }_F$; whence ${\iota }_G(s)={\iota }_G(q)+{\iota }_G(s-q)\le \phi (x)+\phi (x^{\mathrm{\prime }})$; as $s$ was arbitrary and $\phi (x+x^{\mathrm{\prime }})=\sup L_{x+x^{\mathrm{\prime }}}$, the least upper bound is $\le \phi (x)+\phi (x^{\mathrm{\prime }})$ [L8], i.e. $\phi (x+x^{\mathrm{\prime }})\le \phi (x)+\phi (x^{\mathrm{\prime }})$.

For $x,x^{\mathrm{\prime }}>0$ and positive rationals $q,q^{\mathrm{\prime }}$ with ${\iota }_F(q)<x$, ${\iota }_F(q^{\mathrm{\prime }})<x^{\mathrm{\prime }}$, multiplying positives gives ${\iota }_F(q{q}^{\mathrm{\prime }})<x{x}^{\mathrm{\prime }}$, so ${\iota }_G(q){\iota }_G(q^{\mathrm{\prime }})={\iota }_G(q{q}^{\mathrm{\prime }})\in L_{x{x}^{\mathrm{\prime }}}$ and $\phi (x{x}^{\mathrm{\prime }})\ge {\iota }_G(q){\iota }_G(q^{\mathrm{\prime }})$; since positive rationals below $x$ are cofinal (as $x>0$) their images have supremum $\phi (x)>0$, so scaling by ${\iota }_G(q^{\mathrm{\prime }})>0$ and taking sups over $q$ then $q^{\mathrm{\prime }}$ gives $\phi (x{x}^{\mathrm{\prime }})\ge \phi (x)\phi (x^{\mathrm{\prime }})$ by the least-upper-bound calculus [L8].

For $x,x^{\mathrm{\prime }}>0$ and any positive rational $s$ with ${\iota }_F(s)<x{x}^{\mathrm{\prime }}$ we have ${\iota }_F(s)(x^{\mathrm{\prime }}{)}^{-1}<x$; density [L3] gives a rational $q$ with ${\iota }_F(s)(x^{\mathrm{\prime }}{)}^{-1}<{\iota }_F(q)<x$, where ${\iota }_F(s)(x^{\mathrm{\prime }}{)}^{-1}>0$ (as $s>0$, $x^{\mathrm{\prime }}>0$), so ${\iota }_F(q)>0$ and $q>0$; the left inequality gives ${\iota }_F(s)<{\iota }_F(q)x^{\mathrm{\prime }}$, hence ${\iota }_F(s\mathrm{/}q)={\iota }_F(s){\iota }_F(q{)}^{-1}<x^{\mathrm{\prime }}$ (dividing by ${\iota }_F(q)>0$), while ${\iota }_F(q)<x$; therefore ${\iota }_G(s)={\iota }_G(q){\iota }_G(s\mathrm{/}q)\le \phi (x)\phi (x^{\mathrm{\prime }})$, and as the positive rationals $s$ with ${\iota }_F(s)<x{x}^{\mathrm{\prime }}$ are cofinal, $\phi (x{x}^{\mathrm{\prime }})=\sup L_{x{x}^{\mathrm{\prime }}}\le \phi (x)\phi (x^{\mathrm{\prime }})$ by [L8].

Combining the two inequalities, $\phi (x+x^{\mathrm{\prime }})=\phi (x)+\phi (x^{\mathrm{\prime }})$ for all $x,x^{\mathrm{\prime }}\in F$; in particular $\phi (0_F)=0_G$ and $\phi (-x)=-\phi (x)$.

Combining the two inequalities, $\phi (x{x}^{\mathrm{\prime }})=\phi (x)\phi (x^{\mathrm{\prime }})$ whenever $x,x^{\mathrm{\prime }}>0$.

For arbitrary signs, $\phi (0_F\cdot x^{\mathrm{\prime }})=0_G=0_G\cdot \phi (x^{\mathrm{\prime }})$, and for $x<0<x^{\mathrm{\prime }}$ we get $\phi (x{x}^{\mathrm{\prime }})=\phi (-((-x)x^{\mathrm{\prime }}))=-\phi ((-x)x^{\mathrm{\prime }})=-\phi (-x)\phi (x^{\mathrm{\prime }})=\phi (x)\phi (x^{\mathrm{\prime }})$ using step 3.1 and step 3.2; the remaining sign cases are identical, so $\phi (x{x}^{\mathrm{\prime }})=\phi (x)\phi (x^{\mathrm{\prime }})$ for all $x,x^{\mathrm{\prime }}\in F$.

By step 3.1, step 4.1, and $\phi (1_F)={\iota }_G(1)=1_G$ from step 1.3, $\phi$ preserves $+$, $\cdot$, and $1$, so $\phi$ is a field homomorphism $F\to G$.

Hence by [L5] (as $F$ is complete) $\phi$ is injective and order-preserving in both directions, and by [L4] it fixes $\mathbb{Q}$: $\phi \circ {\iota }_F={\iota }_G$.

The construction and steps 1.1-6.1 used only that $F$ and $G$ are complete ordered fields with canonical embeddings ${\iota }_F,{\iota }_G$; applying that entire argument verbatim with the roles of $F$ and $G$ interchanged shows the symmetric map $\psi :G\to F$ is likewise an injective, order-preserving field homomorphism that fixes $\mathbb{Q}$.

For $x\in F$, since $\phi$ fixes $\mathbb{Q}$ and is order-preserving in both directions, ${\iota }_G(q)<\phi (x)⟺{\iota }_F(q)<x$, so $\psi (\phi (x))=\sup \{{\iota }_F(q):{\iota }_F(q)<x\}=x$ by density [L3]; symmetrically $\phi (\psi (y))=y$, so $\phi$ is a bijection with inverse $\psi$.

Thus $\phi$ is a bijective field homomorphism order-preserving in both directions, i.e. an ordered-field isomorphism $F\cong G$ fixing $\mathbb{Q}$.

For uniqueness let $\chi :F\to G$ be any ordered-field isomorphism; being such it is in particular a field homomorphism ([L7]), so by [L4] it fixes $\mathbb{Q}$, and it is order-preserving, so for each $x$ every ${\iota }_G(q)\in L_x$ equals $\chi ({\iota }_F(q))<\chi (x)$, making $\chi (x)$ an upper bound of $L_x$, hence $\chi (x)\ge \phi (x)$.

Conversely, were $\chi (x)>\phi (x)$, density [L3] would give a rational $q$ with $\phi (x)<{\iota }_G(q)<\chi (x)$; then ${\iota }_G(q)>\sup L_x$ forces ${\iota }_F(q)\ge x$, since ${\iota }_F(q)<x$ would put ${\iota }_G(q)$ into $L_x$ and hence below $\sup L_x$; so $\chi (x)\le \chi ({\iota }_F(q))={\iota }_G(q)<\chi (x)$, which is impossible, hence $\chi (x)\le \phi (x)$.

Therefore $\chi (x)=\phi (x)$ for every $x$, so $\chi =\phi$: the ordered-field isomorphism is unique.

Applying this to any two constructions of $\mathbb{R}$, which are complete ordered fields, $\mathbb{R}$ is the unique complete ordered field up to a unique ordered-field isomorphism, and ${\iota }_F:\mathbb{Q}\to \mathbb{R}$ exhibits $\mathbb{Q}$ as an ordered subfield.