Nearly every mathematician nowadays is familiar with the fact that
there is up to isomorphism only one complete ordered field, the
real numbers.
Theorem. Any two complete ordered fields are isomorphic.
Proof. $\newcommand\Q{\mathbb{Q}}\newcommand\R{\mathbb{R}}$Let us observe first that every complete ordered field $R$ is
Archimedean, which means that there is no number in $R$ that is
larger than every finite sum $1+1+\cdots+1$. If there were such a
number, then by completeness, there would have to be a least such
upper bound $b$ to these sums; but $b-1$ would also be an upper
bound, which is a contradiction. So every complete ordered field is
Archimedean.
Suppose now that we have two complete ordered fields, $\R_0$ and
$\R_1$. We form their respective prime subfields, that is, their
copies of the rational numbers $\Q_0$ and $\Q_1$, by computing
inside them all the finite quotients
$\pm(1+1+\cdots+1)/(1+\cdots+1)$. This fractional representation
itself provides an isomorphism of $\Q_0$ with $\Q_1$, indicated
below with blue dots and arrows:
Next, by the Archimedean property, every number $x\in\R_0$
determines a cut in $\Q_0$, indicated in yellow, and since $\R_1$
is complete, there is a counterpart $\bar x\in\R_1$ filling the
corresponding cut in $\Q_1$, indicated in violet. Thus, we have
defined a map $\pi:x\mapsto\bar x$ from $\R_0$ to $\R_1$. This map
is surjective, since every $y\in\R_1$ determines a cut in $\Q_1$,
and by the completeness of $\R_0$, there is an $x\in\R_0$ filling
the corresponding cut. Finally, the map $\pi$ is a field
isomorphism since it is the continuous extension to $\R_0$ of the
isomorphism of $\Q_0$ with $\Q_1$. $\Box$
My expectation is that this theorem is familiar to almost every
contemporary mathematician, and I furthermore find this theorem
central to contemporary mathematical views on the philosophy of
structuralism in mathematics. The view is that we are entitled to
refer to the real numbers because we have a categorical
characterization of them in the theorem. We needn't point to some
canonical structure, like a canonical meter-bar held in some
special case deep in Paris, but rather, we can describe the
features that make the real numbers what they are: they are a
complete ordered field.
Question. Who first proved or even stated this theorem?
It seems that Hilbert would be a natural candidate, and I would
welcome evidence in favor of that. It seems however that Hilbert
provided axioms for the real field that it was an Archimedean
complete ordered field, which is strangely redundant, and it isn't
clear to me whether he actually had the categoricity result.
Did Dedekind know it? Or someone else? Please provide evidence; it
would be very welcome.
Best Answer
Joel, I believe this was first explicitly stated and proved by E.V. Huntington in his classic paper: COMPLETE SETS OF POSTULATES FOR THE THEORY OF REAL QUANTITIES, Trans. Am. Math. Soc. vol. 4, No. 3 (1903), pp. 358-370. See Theorem II', p. 368.
Edit (June 14, 2020): It is perhaps worth adding that in 1904, the year following the publication of Huntington's paper, O. Veblen published his paper A System of Axioms for Geometry, Trans. Am. Math. Soc. vol. 5, no. 3, pp. 343-384, in which he introduced the idea of a categorical system of axioms. He illustrated his conception with Huntington's above mentioned characterization of the reals (pp. 347-348). No doubt, this is mentioned in the paper referred to below by Ali Enayat.