# History of the construction of $\mathbb{R}$

When did the constructions of Reals take place? What is the latest construction the one due to Cantor (by Cauchy Sequences) or the Dedekind? I ask because the trustful reference (baby Rudin) that I looked, tells they were published in the same year 1872.

That seems to me, Cantor's idea was more fruitful (however was quite similar) because it was used to complete any Metric space. Nevertheless the Dedekind construction may rest as an Historical thing when the Cantor idea should be the one taught obligatorily in the Analysis courses.

Personal commentary: I don't think the Dedekind's idea can be used in any other construction.

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Did you check the wikipedia article on real numbers before asking? – Harald Hanche-Olsen Feb 19 '12 at 20:21
The article is wrong Cantor and Dedekind gave the construction of the Reals in 1872. See Rudin, Principles of Real Analysis. – checkmath Feb 19 '12 at 21:58
@chessmath: To avoid further downvotes, closing votes, and to attract better answers you might want to consider editing the question to add your previous comment; what you have read on the topic and so forth. Regarding the latest construction, I can tell you that our freshman year just finished constructing it last semester. The model they built hasn't been used yet! – Asaf Karagila Feb 20 '12 at 0:11

To complement the fine answers given earlier, I would just point out that the first construction of the reals was arguably given considerably earlier than the above discussion suggests. Namely, Simon Stevin already envisioned representing each number by an unending decimal, and emphasized that there is no difference between rational numbers and other numbers (such as surds) in this respect. Arguably, the reason mathematicians like Cauchy did not bother "constructing" the real numbers is because they felt such a number system had already been constructed, and used for centuries. More details can be found in this article on Stevin.

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What follows is a LaTeX rendering of a sci.math post I made in 17 June 2006 that may be of use in connection with the answers that already appear.

There are many ways to construct the real numbers. Among the more common methods are the use of Cauchy sequences, Dedekind cuts, decimal expansions, and nested intervals with rational endpoints. For some other ways to construct the real numbers, see Dhombres [1], Knopfmacher/Knopfmacher [4] [5] [6], Maier/Maier [7], Rieger [10], and Shiu [11]. For historical issues related to various constructions of the real numbers, see Ferreirós [2] (Chapter IV), Manheim [8] (Sections 4.6-4.13, pp. 76-95), Pellicer [9], and Simsa [12]. Fowler [3] proves $\sqrt{2} \times \sqrt{3} = \sqrt{6}$ for several ways of constructing the real numbers, after a brief discussion about the significance of this identity.

[1] Jean G. Dhombres, Real numbers from Cauchy to Robinson, Southeast Asian Bulletin of Mathematics 1 (1977), 9-20. [MR 58 #21199]

[2] José Ferreirós, Labyrinth of Thought. A History of Set Theory and Its Role in Modern Mathematics, Science Networks / Historical Studies #23, Birkhäuser Verlag, 1999, xxi + 440 pages. [MR 2000m:03005; Zbl 934.03058]

[3] David Fowler, Dedekind's theorem: $\sqrt{2} \times \sqrt{3} = \sqrt{6},$ American Mathematical Monthly 99 #8 (October 1992), 725-733. [MR 93h:01022; Zbl 766.01016]

[4] Arnold Knopfmacher and John Knopfmacher, A new construction of the real numbers (via infinite products), Nieuw Archief voor Wiskunde (4) 5 (1987), 19-31. [MR 88i:11007; Zbl 624.10007]

[5] Arnold Knopfmacher and John Knopfmacher, Two concrete new constructions of the real numbers, Rocky Mountain Journal of Mathematics 18 (1988), 813-824. [MR 90k:26003a; Zbl 677.10006]

[6] Arnold Knopfmacher and John Knopfmacher, Two constructions of the real numbers via alternating series, International Journal of Mathematics and Mathematical Sciences 12 (1989), 603-613. [MR 90k:26003b; Zbl 683.10008]

[7] David E. Maier and Eugene A. Maier, Construction of the real numbers, (Two-Year) College Mathematics Journal 4 #1 (Winter 1973), 31-35.

[8] Jerome H. Manheim, The Genesis of Point Set Topology, Pergamon Press, 1964, xiii + 166 pages. [MR 37 #2561; Zbl 119.17702]

[9] Manuel López Pellicer, Las construcciones de los números reales [Constructions of real numbers], pp. 11-33 in Historia de la Mathemática en el Siglo XIX (Parte 2), Real Academia de Ciencias Exactas, Físicas y Naturales (Madrid), 1994. [MR 98f:01030; Zbl 952.00024]

[10] Georg Johann Rieger, A new approach to the real numbers (motivated by continued fractions), Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft 33 (1982), 205-217. [MR 84j:26002; Zbl 513.10009]

[11] Peter Shiu, A new construction of the real numbers, Mathematical Gazette 58 #403 (March 1974), 39-46.

[12] Jaromír Simsa, Development of the concept of real numbers (Czech), pp. 259-282 in Mathematics in the 16th AND 17th Centuries (Czech) (Jevícko, 1997), Dej. Mat./Hist. Math. #12, Prometheus, Prague, 1999. [MR 2003g:01001]

An additional reference I found in March 2013:

[13] Alexandru Pintilie, A construction without factorization for the real numbers, Libertas Mathematica 8 (1988), 155-158. [MR 90e:00002; Zbl 661.26003]

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This is a great answer! – Pete L. Clark Feb 21 '12 at 16:42

The only reference I can give is Akihiro Kanamori's historical overview in The Higher Infinite, in which he cites papers by Dedekind and Cantor for their constructions - both given in 1872.

To remark on your personal comment, I have to say two things (which are way too long to fit into a comment):

1. The fact that Cantor's construction can be carried out in a general metric space has one caveat, as one could argue that it is simply the same thing in the case of $\mathbb R$. However it is not: in the general case we use the completeness of $\mathbb R$, while when constructing $\mathbb R$ we do not yet know that it is complete. The proof of its completeness is not very hard, but it is something one needs to do nonetheless.

2. Dedekind's construction is actually very useful. If we take a partial order $(P,\le)$ we can embed it densely in the set of its cuts, $L_p=\{x\mid x<p\}$ ordered by inclusion. This gives us a complete partial order, that is every bounded set has a least upper bound.

If we started a separative partial order then the result is a complete Boolean-algebra in which $P$ is embedded densely. The original separative partial order could have been a Boolean-algebra to begin with, which will then yield the completion of a Boolean-algebra.

This quite useful in forcing via Boolean-valued models, and it can also help us transfer a problem from partially ordered sets into Boolean-algebras.

Indeed the Dedekind construction has its own uses. You may have said what you said simply because one do not run into Dedekind completions very often outside rather set-theoretical settings (not just set theory, but fields close to it) which is not often appearing in undergrad or even grad level studies.

It is also worth noting that from Dedekind's construction it is a lot easier to prove that every bounded subset of $\mathbb R$ has a supremum, and it is very easy to find this supremum as well.

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I sometimes wonder whether the "undervaluation" of order theory (lattice theory) doesn't cause more "real world harm" than the mathematical community is willing to admit. The operations "min" and "max" (or "inf" and "sup", or "meet" and "join") have similar nice properties like addition and multiplication, and occur frequently in "normal" computer programs. Yet they lack first class citizen support in many programing languages. Parallel constructs omit reduce operations for "min" and "max", parentheses are required even so "min" and "max" are associative, clunky predefined min/max macros... – Thomas Klimpel Feb 21 '12 at 0:13
@Thomas: I don't know if many mathematicians would care for this, I have also learned in my few years as a student that computer scientists and programmers are two very different creatures. In particular, dealing with sets and array is harder than manipulating numbers. Now if there was a processor running an architecture based on set theory... that's something I would like to see! – Asaf Karagila Feb 21 '12 at 0:29
There are many finite algebraic structures (take finite fields as an example) one might want to "use" in a computer program. I noticed that the hardware (=processor) itself often offers good support for this (as if hardware engineers had foreseen such use cases), but many high level programming languages force you to go through hoops. So it's no coincidence that the sse2/sse3 instructions set directly supports "min" and "max" operations, but high level programming languages don't. – Thomas Klimpel Feb 21 '12 at 8:36
@Thomas: I believe that SSE2/3/... were introduced in this millennium, which makes them a far newer technology. Ask yourself why won't all cars have ABS? Well, some were simply manufactured so long ago that it's moot to install such system on such old car. – Asaf Karagila Feb 21 '12 at 8:43
@ThomasKlimpel: The "tropical semiring" and tropical geometry put the min operation front and center. It's a fascinating area. – Grumpy Parsnip Feb 21 '12 at 17:32

As to what is the most recent construction of the reals, I can't say for sure. I think it is not too fashionable to come with new constructions, but one can construct more general number systems that contain the reals as a subset‚ without constructing the reals first. One nice and fairly recent example is Conway's surreal numbers, first constructed around 1970, I think. Donald Knuth's book Surreal numbers is a quite readable exposition, a bit like a mystery novel.

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Here are some highlights, concerning mostly the irrationals, from chapter 41 of Morris Kline's Mathematical Thought from Ancient to Modern Times.

Euclid had a notion of "incommensurable ratios", which Kline argues are just the irrationals from a different point of view. Euclid also had the notion of defining equality of incommensurable ratios by, given one of these ratios, dividing the rational numbers into two classes, those for which the rational is less than the incommensurable ratio, and those which are greater. This reminds one of Dedekind cuts; a fact which Dedekind himself acknowledged.

William R. Hamilton offered the first (incomplete) treatment of irrational numbers in two papers read before the Royal Irish Academy in 1833 and 1835. He also had a notion of Dedekind cuts.

Cantor pointed out that the previous work tried to define the irrationals as limits of rationals, whilst the limit, if irrational, is not defined logically unless the irrationals are already defined. At this time, 1859, Weierstrass gave a theory of the irrationals. This was supposedly published in Die Elemente der Arithmetik in 1872 by H. Kossack; though Weierstrass disowned the work

In 1869 Charles Méray gave a definition of the irrationals based on the rationals.

George Cantor gave his theory in 1871.

Eduard Heine gave his theory in 1872 in the Journal für Mathematik (74, 172-178).

In the same year Dedekind gave his theory in Stetigkeit und irrationale Zahlen (3, 314-334).)

After all this, the rational numbers were put on a rigorous basis, starting with the integers, with works of Dedekind in his Was sind und was sollendie Zahlen(1888, 16, 335-391) and, more notably by Peano with his axiomatic approach in 1889 in his Arithmetices Principia Nova Methodo Exposita.

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