Probably what Abbott in Understanding Analysis calls the axiom of completeness: every set that is bounded above has a least upper bound.
Making this stipulation distinguishes the reals from the rationals, as e.g. the set of numbers whose square is less than 2 is bounded above by any number whose square is greater than or equal to two, but among the rationals there is no least upper bound: given any rational number whose square is greater than or equal to two we can always find a smaller such rational.
Assuming the axiom of completeness, we define the square root of two as the least upper bound of the set of numbers whose square is less than two
But that is an axiom, not a construction! The point of Dedekind cuts is that they give a construction of the real numbers, and one can prove that this satisfies the Axiom of Least Upper Bounds.
You don't need a construction for a calculus class. If you do need one Cauchy sequence completion is more generalizable and somewhat easier to work with.
I don’t really know what a “calculus” class is since here (the UK) that term isn’t really used for university-level mathematics; we’d usually say “analysis” instead, but I know that “analysis” is a class in the US too, so I don’t know if calculus is closer to what we would do just prior to university (a bit of limits, differentiation, Riemann integrals, a bit of vector calculus).
Virtually every first year UK undergraduate analysis course will start with a construction of the reals via Dedekind cuts, and this is about the level that this book is pitched at.
The original commenter suggested that “least upper bounds” is a simpler approach, and that Hardy’s book is outdated by using Dedekind cuts; it may be that constructing the reals is not something that would be done at “calculus”-level in the US, but clearly the book isn’t aimed at that level.
Dedekind cuts (or Cauchy sequences) are totally standard, and I don’t think it’s fair to criticise their use at all.
In the U.S., there is typically a separation between calculus and real analysis. Though, the amount of difference between the two depends on the university.
In calculus, there is more emphasis on learning how to mechanically manipulate derivatives and integrals and their use in science and engineering. While this includes some instruction on proving results necessary for formally defining derivatives and integrals, it is generally not the primary focus. Meaning, things like limits will be explained and then used to construct derivatives and integrals, but the construction of the reals is less common in this course. Commonly, calculus 1 focuses more on derivatives, 2 on integrals, and 3 on multivariable. However, to be clear, there is a huge variety in what is taught in calculus and how proof based it is. It depends on the department.
Real analysis focuses purely on proving the results used in calculus classes and would include a discussion on the construction of the reals. A typical book for this would be something like Principles of Mathematical Analysis by Rudin.
I'm not writing this because I don't think you don't know what these topics are, but to help explain some of the differences between the U.S. and elsewhere. I've worked at universities both in the U.S. and in Europe and it's always a bit different. As to why or what's better, no idea. But, now you know.
Side note, the U.S. also has a separate degree for math education, which I've not seen elsewhere. No idea why, but it also surprised me when I found out.
There's a POV that learning math and learning how to teach math effectively are two orthogonal things.
If one only took the method of teaching that is most common in US university lecture halls, and applied it to a small class of pre-teens or teenagers, it probably wouldn't be very effective.
I went to a University of California school which had 3 calculus tracks - one for life/social sciences students (eg biology, econ), one for physical sciences (chemistry, physics, math, ...), and an honors track.
High school went up through what we call Algebra II. Calculus is an Advanced Placement (AP) course that most students don't take.
I took physical sciences calc + multivariate calc (1 year including summer), an intro to proofs and set theory course, and then finally a rigorous construction of reals was taught in our upper division real analysis course. So somewhere in my second year as a math major. Though I had already researched the constructions myself out of curiosity.
Apart from the material being extraneous for anyone outside the major, I think they were in a sense trying to be more rigorous by first requiring set theory which included constructions of the integer and rational number systems.
Where we define the real numbers as the least upper bounds of special sets. There is a bijection between these sets and the set of real numbers which we commonly think of and that bijection is the least upper bound of such sets.
I haven't looked at Hardy's but the presentation in Spivak is also Dedekind cuts. Perhaps Hardy uses a different approach and OP misnamed it? Rudin's chapter 1 annex also use Dedekind's cuts.
It looks like Hardy used Dedekind cuts from starting with the second edition (1914), but not in the first edition (1908).
What's the advantage of Dedekind cuts over say equivalence classes of Cauchy sequences of rational numbers? Particularly if you start out by introducing the integers and rational numbers as equivalence classes as well.
The equivalence class of Cauchy sequences is vastly larger and misleading compared to those of integers and rational numbers. You can take any finite sequence and prepend it to a Cauchy sequence and it will represent the same real number. For example, a sequence of 0,0,0,...,0 where the number of dots is the count of all the atoms in the universe and then followed by the decimal approximations of pi: 3, 3.1, 3.14, 3.141, ... The key component is the error clause of getting close, but that can vary greatly from sequence to sequence as to when that happens. The cute idea of being able to look at a sequence and see roughly where it is converging just is not captured well in the reality of the equivalence classes.
More or less, one can think of a Cauchy sequence of generating intervals that contain the real number, but it can be arbitrarily long before the sequence gets to "small" intervals. So comparing two Cauchy sequences could be quite difficult. Contrast that with the rational numbers where a/b ~ c/d if and only if ad = bc. This is a relatively simple thing to check if a, b, c, and d are comfortably within the realm of human computation.
Dedekind cuts avoid this as there is just one object and it is assumed to be completed. This is unrealistic in general though the n-roots are wonderful examples to think it is all okay and explicit. But if one considers e, it becomes clear that one has to do an approximation to get bounds on what is in the lower cut. The (lower) Dedekind cut can be thought of as being the set of lower endpoints of intervals that contain the real number.
My preference is to define real numbers as the set of inclusive rational intervals that contain the real number. That is a bit circular, of course, so one has to come up with properties that say when a set of intervals satisfies being a real number. The key property is based on the idea behind the intermediate value theorem, namely, given an interval containing the real number, any number in the interval divides the interval in two pieces, one which is in the set and the other is not (if the number chosen "is" the real number, then both pieces are in the set).
There is a version of this idea which is theoretically complete and uses Dedekind cuts to establish its correctness[1] and there is a version of this idea which uses what I call oracles that gets into the practical messiness of not being able to fully present a real number in practice[2].
> The equivalence class of Cauchy sequences is vastly larger and misleading compared to those of integers and rational numbers. You can take any finite sequence and prepend it to a Cauchy sequence and it will represent the same real number. ...
This can be addressed practically enough by introducing the notion of a 'modulus of convergence'.
> The equivalence class of Cauchy sequences is vastly larger and misleading compared to those of integers and rational numbers. You can take any finite sequence and prepend it to a Cauchy sequence and it will represent the same real number.
What's the misleading part of this supposed to be?
The equivalence classes of integers: pairs of naturals with (a, b) ~ (c, d) := (a + d) = (b + c).
The equivalence classes of rationals: pairs of integers with (a, b) ~ (c, d) := ad = bc.
It’s “easy” to tell whether two integers/rationals are equivalent, because the equivalence rule only requires you to determine whether one pair is a translation/multiple resp. of the other (proof is left to the reader).
Cauchy sequences, on the other hand, require you to consider the limit of an infinite sequence; as the GP points out, two sequences with the same limit may differ by an arbitrarily large prefix, which makes them “hard” to compare.
We can formalise this notion by pointing out that equality of integers and rationals is decidable, whereas equality of Cauchy reals is not. On the other hand, equality of Dedekind reals isn’t decidable either, so it’s not that Cauchy reals are necessarily easier than Dedekind reals, but more that they might lull one into a false sense of security because one might naively believe that it’s easy to tell if two sequences have the same limit.
The intuition of a sequence is that the terms get closer to the convergence point. Looking at the first trillion elements of a sequence feels like it ought to give one some kind of information about the number. But without the convergence information, those first trillion elements of the sequence can be wholly useless and simply randomly chosen rational numbers. This is an "of course", but when talking about defining a real number with these sequences, as opposed to approximating them, this gives me a great deal of unease.
In particular, it is quite possible to prove a theorem that a sequence is Cauchy, but that there is no way to explicitly figure out N for a given epsilon. The sequence is effectively useless. This presumably is possible, and common, with using the Axiom of Choice. One can even imagine an algorithm for such a sequence that produces numbers and eventually converges, but the convergence is not knowable. Again, if this is just approximating something, then we can simply say it is a useless approximation scheme. But defining real numbers as the equivalence class of Cauchy sequences suggests taking such a sequence seriously in some sense and is the answer.
In contrast, consider integer and rational number versions, it is quite immediate how to reduce them to their canonical form, assuming unlimited finite arithmetic ability. For example, 200/300 ~ 2/3 and one recognizes that 200/300 and 2/3 are different forms of what we take to be the same object for most of our purposes. There is no canonical Cauchy sequence to reduce to and concluding two sequences are equivalent could take a potentially infinite number of computations /comparisons. While that is somewhat inherent to the complexity of real numbers, it feels particularly acute when it is something that must be done in defining the object.
Dedekind cuts have the opposite problem. There is only one of them for an irrational number, but it is not entirely clear what we would be computing out as an approximation, particularly if the lower cut viewpoint is adopted.
Intervals, on the other hand, inherently contain the approximation information. By dividing them and picking out the next subinterval, one also has a method to computing out a sequence of ever better approximations. I suppose one could prove the existence of the family of containment intervals without explicitly being able to produce them, but at least the emptiness of the statement would be quite clear (nothing is produced) in contrast to the sequences that could produce essentially meaningless numbers for an arbitrarily large number of terms.
Making this stipulation distinguishes the reals from the rationals, as e.g. the set of numbers whose square is less than 2 is bounded above by any number whose square is greater than or equal to two, but among the rationals there is no least upper bound: given any rational number whose square is greater than or equal to two we can always find a smaller such rational.
Assuming the axiom of completeness, we define the square root of two as the least upper bound of the set of numbers whose square is less than two