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:

categoricity of reals as complete ordered field

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.

  • 8
    $\begingroup$ This is a great question, but, like all questions about history of science and mathematics, it seems like it should go on HSM. (The standard response is that that site is less active, and at least part of that is because there's so much HSM activity here!) $\endgroup$ – LSpice Jun 13 '20 at 21:21
  • 12
    $\begingroup$ @LSpice The standard response is really that the quality of HSM is awful and many experts and interested scholars avoid it as a result. Sending a question there is usually a disservice. $\endgroup$ – Andrés E. Caicedo Jun 13 '20 at 21:46
  • 7
    $\begingroup$ @AndrésE.Caicedo, right, but it's circular: HSM will never get any better if the most qualified people keep avoiding it, so those who are interested in HSM should ask and answer questions there, not re-purpose MO for it. We are very strict with new users that MO is only for research-level questions in mathematics; we should be equally strict amongst ourselves. $\endgroup$ – LSpice Jun 13 '20 at 22:38
  • 14
    $\begingroup$ @LSpice I think your understanding of what this site is is perhaps too narrow. I feel the question belongs here. $\endgroup$ – Andrés E. Caicedo Jun 14 '20 at 4:11
  • 16
    $\begingroup$ @LSpice One can turn your argument around: it is precisely the excessive efforts to limit interesting questions on MO that has led in recent years to a reduced level of quality and engagement with MO. Have you observed this? To the extent that you are successful in transferring mathematically interesting questions to another site, I would argue that you are working towards the decline of MO. Let's have the interesting questions here! My MO policy has always been: questions are welcome on MO if they are of interest to research-level mathematicians. $\endgroup$ – Joel David Hamkins Jun 14 '20 at 18:20

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.

  • 2
    $\begingroup$ This paper can also be accessed at ams.org/tran/1903-004-03/S0002-9947-1903-1500647-9/…. $\endgroup$ – shane.orourke Jun 13 '20 at 21:44
  • 5
    $\begingroup$ Hutington's work was based on previous work by Dedekind and Hilbert:, but nevertheless, the following article corroborates Ehrlich's claim that E.V. Huntington can be credited to be the first to have formulated and proved the categoricity of the second order theory of the real field (in modern parlance). Completeness and Categoricity. Part I: Nineteenth-century Axiomatics to Twentieth-century Metalogic, by S. Awodey & E. H. Reck (a preprint can be found via andrew.cmu.edu/user/awodey/preprints/cc/ccI.pdf). It appeared in print in History and Philosophy of Logic, 23: 1–30 (2002). $\endgroup$ – Ali Enayat Jun 14 '20 at 17:02
  • $\begingroup$ What about Hoelder's paper from 1901: 'Der Quantitat und die Lehre yom Mass', in Berichte uber die Verhandlungen der Koniglich Siichsuschen Gesellschaft der Wissenschaften zu Leipzig, Matematisch-Physische Classe, 1-64. $\endgroup$ – Zvonimir Sikic Jun 17 '20 at 11:59
  • $\begingroup$ It is proved there that Archemedean systems of magnitudes with no minimal magnitudes, are isomorphicaly and densely embeddable in R+ (if they have minima they are isomorphic to Z+). If a system of magnitudes is continuous, i.e. does not have empty Dedekind cuts, then it is isomorphic to R+. Systems of magnitudes is a linearly ordered semigroup with restricted difference. $\endgroup$ – Zvonimir Sikic Jun 17 '20 at 12:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.