Is the real number structure unique?

Question

For a frame of reference, I'm an undergraduate in mathematics who has taken the introductory analysis series and the graduate level algebra sequence at my university.

In my analysis class, our book lists axioms describing the structure of the reals. This seemed unnatural to me, as we can often write definitions or lists of rules that no set actually satisfies. So the axiomatic approach doesn't reassure me that the thing we are discussing can actually exist.

Our teacher talked to us about dedekind cuts as a way of explicitly constructing reals, which seemed more useful to me.

In my modern algebra class we discussed completion of the rationals as a way to construct the real numbers.

But this leads me to wonder - How do we know that all of these approaches and constructions result in the same structure, namely $\mathbb{R}$? Also, are the real numbers the only complete, totally ordered field? If so, why?

Answer 1

To paint a more complete picture, you are right in that an axiomatization may very well have no model. An axiomatization is meaningless if nothing satisfies it. But if we can prove that there is a model, and the axioms are the only properties we care about, then we can happily work within the axiomatization to prove theorems about the structure that we have axiomatized. That is precisely what we are doing when we use an axiomatization of the real numbers to prove theorems about real numbers.

Now there are two common axiomatizations of the reals. One is a second-order theory including the completeness axiom (every subset of the reals with an upper bound has a least upper bound). By itself, this axiom is useless, because there is no axiom that asserts the existence of any set of reals at all! However, when we use this second-order theory we are always working outside the theory in the foundational system (such as ZFC set theory) where we do have axioms that allow construction of subsets of the reals. It is this particular axiomatization that has a unique model; there is a unique model of the second-order theory of the reals up to isomorphism. That implies immediately that all structures that you construct (like the Cauchy completion of the rationals or the Dedekind completion of the rationals) that satisfies this second-order axiomatization must be isomorphic to one another.

The core reason behind the uniqueness of complete ordered field is that any two such fields must contain an isomorphic copy of the rationals, and every element in each field cuts the rationals in the field into two parts, and the lower part has a least upper bound, and that different elements cut the rationals in different ways (by the Archimedean property that follows from the completeness property as well). This gives a one-to-one correspondence between the reals in one field to the reals in the other field. One can say that it is the rigidity and denseness of the rationals that is the key.

However, the other common axiomatization of the reals is the theory of real closed fields. Note that this axiomatization does not have a unique model. The computable reals form a countable real closed field, and satisfies every first-order sentence that $\mathbb{R}$ satisfies. It may be instructive to see the same phenomenon with the natural numbers, which form the unique model of the second-order Peano's axioms (which was his original formulation) up to isomorphism, while there are many non-isomorphic models of first-order PA. The distinction here between the first-order induction schema and the single second-order induction axiom must be appreciated for one to understand how the second-order theory can pin down the natural numbers unlike the first-order one. More specifically, second-order induction applies to every subset of the natural numbers (as seen from the foundational system), while first-order induction applies only to subsets that can be described using an arithmetical formula. There are uncountably many subsets, but only countably many formulae.

Answer 2

The (fake¹) history of mathematics is

• We believed we understood the real numbers very well
• We wrote down simple properties (we believed) that the real numbers satisfy
• We checked that these properties are enough to actually prove from axioms all of the things we were proving about real numbers

Uniqueness, incidentally, comes from the third point. For example, one of those "things we were proving about real numbers" is that they are the completion of the rationals. Since the rationals are unique and completions are unique, the completion of the rationals must also be unique.

(that a complete ordered field contains the rationals follows from "field" and "ordered", and the topology on a complete ordered field follows from "ordered")

As usual, by "unique" I really mean "unique up to isomorphism". In this case, we even have "unique up to unique isomorphism".

_{1: true history is strange; it may not have happened in this order, or precisely like this}

Answer 3

A lot of books devoted to the development of the real number system (and other number systems) were published in the 1960s. During the past few years I've been keeping a list of such books (regardless of when they were published) when I happen to see one in a library, and the ones I've come across so far are listed below.

Note that if we expand the list to books in real analysis, "transition to advanced mathematics", abstract algebra, metric spaces and/or topology, etc. that include such a treatment, the list of such books would be at least several hundred. Thus, I've restricted the list below to only those books that are primarily devoted to this topic. The links I've provided are not necessarily to the same edition that I've given information about.

Besides what others have recommended, you can try visiting a nearby college or university library and look for some of these books. Some of these books will probably be more useful to you than others for what you want, and you will likely find other books in the same general shelving location that are not in the list below.

[1] Leon Warren Cohen and Gertrude Ehrlich, The Structure of the Real Number System, The University Series in Undergraduate Mathematics, D. Van Nostrand Company, 1963, viii + 116 pages.

[2] Solomon Feferman, The Number Systems. Foundations of Algebra and Analysis, Addison-Wesley Publishing Company, 1964, xii + 418 pages.

The 2nd edition was published by Chelsea Publishing Company in 1989 (xii + 418 pages).

[3] Norman Tyson Hamilton and Joseph Landin, Set Theory and the Structure of Arithmetic, Allyn and Bacon, 1961, xii + 264 pages.

[4] Edmund Jecheksel Landau, Foundations of Analysis, 1951, Chelsea Publishing Company, 1951, xiv + 134 pages.

Translation by Fritz Robert Steinhardt of the 1930 German edition (xiv + 134 pages).

[5] Elliott Mendelson, Number Systems and the Foundations of Analysis, Academic Press, 1973, xii + 358 pages.

Reprinted by Robert E. Krieger Publishing Company in 1985 (xii + 358 pages). Reprinted by Dover Publications in 2008 (xii + 308 pages).

[6] John Meigs Hubbell Olmsted, The Real Number System, Appleton-Century Monographs in Mathematics, Appleton-Century-Crofts, 1962, xii + 216 pages.

[7] Francis Dunbar Parker, The Structure of Number Systems, Teachers. Mathematics Reference Series, Prentice-Hall, 1966, xiv + 137 pages.

[8] Joseph [Joe] Buffington Roberts, The Real Number System in an Algebraic Setting, A Series of Undergraduate Books in Mathematics, W. H. Freeman and Company, 1962, x + 145 pages.

[9] Hugh Ansfrid Thurston, The Number-System, Interscience Publishers, 1956, viii + 134 pages.

Reprinted (slightly corrected) by Dover Publications in 1967 and 2007.

(ADDED 3 YEARS LATER)

See also the 12 related items (mostly papers) given in this 17 June 2006 sci.math post (an additional item is in this 18 June 2006 sci.math post). I'm including these in my answer because the link I gave in a comment 3 years ago no longer works.

Answer 4

As Pugh states in Real Mathematical Analysis, the real numbers are the unique, up to order-preserving isomorphism, complete ordered field containing the rational numbers as an ordered subfield. For suppose $\mathbb{F}$ is a complete ordered field containing $\mathbb{Q}$ as an ordered subfield. Then the map $y \mapsto \{ q \in \mathbb{Q} :q < y \text{ in } \mathbb{F}\}$ is a order-preserving isomorphism from $\mathbb{F}$ to $\mathbb{R}$, where $\mathbb{R}$ is viewed as a set of Dedekind cuts.

Answer 5

A real number system is a tuple $(\mathbb{R},\, +,\, .,\, < )$ which is an ordered field that is order complete (or instead of order complete there are several equivalent properties). Any two such systems are indeed isomorphic, meaning there is a one-to-one correspondence preserving the structure, all this has been already pointed out by others.

What nobody seems to have said and which is a MAIN POINT (pun intended), is that this isomorphism is UNIQUE; there is no choice of which elements correspond.

In particular an isomorphism of $(\mathbb{R},\, +,\, .,\, < )$ with itself must be the identity function taking $x$ to $I(x)=x$. This is very different than say for three dimensional vector spaces, any two have many vector space isomorphisms . This uniqueness explains why the particular constructions (Dedekind cuts, Cauchy sequences, and others) don't matter much to analysis; they do matter to people interested in foundations of almost all of mathematics within set theory from set theory axioms (me included) but not to analysts (me included).

Answer 6

The uniqueness of the real numbers is a bit of a fantasy that reinforces Platonist ideas about mathematics, which may or may not be a bad thing. At any rate, consider the following thought experiment that should indicate that even the uniqueness of the natural numbers $\mathbb{N}$ is dubious:

Is there a set of cardinality strictly between $\mathbb N$ and $\mathbb{N}^{\mathbb N}$?

At a more technical level, it could be pointed out that the so-called categoricity of the real numbers implicitly depends on the choice of the background set theory. Change the background theory and you change the real numbers.

At a further technical level, in Edward Nelson's Internal Set Theory, infinitesimals can be found within the ordinary real numbers. Not every college freshman's idea of the real numbers, is it?

Thus, such "uniqueness" is dubious from at least two points of view: (1) the theory is not unique; and (2) the model is not unique.

Answer 7

No and yes. It depends by what you mean by unique.

First, for the No side. Here are some examples of some questions in which the answer completely dependent on which $\mathbb{R}$ are you talking about (and were significant unsolved problems in analysis for a long time before people realize it cannot be answered with our current axioms):

• Let $f$ be an arbitrary function that assign each real number to a countable subset of real numbers, can you always find 2 numbers $x,y$ such that $x\notin f(y)$ and $y\notin f(x)$.

• Is there a subset of $\mathbb{R}^{2}$ such that every vertical line intersect the set at countably many points and every horizontal line intersect the set at uncountably many points.

Now, for the Yes side. It is a standard theorem in analysis is that any 2 ordered fields with the Dedekind complete property are isomorphic.

How are they both possible? When you talk about unique, you are talking about real number inside the same universe only. It say nothing about real number in a different universe. In particular, 2 "distinct" (e.g. they have different properties) ordered fields with Dedekind complete properties cannot lie in the same universe. However, it is entirely possible that 2 universes contain different real number.

People can't even agree as to which is truly the real number. People have better ideas about what natural number looks like, and even then, we can't even write down axioms that will pin down the natural numbers.