Basics: Real Numbers

What are the real numbers?

Before I go into detail, I need to say up front that I hate the term
real number. It implies that other kinds of numbers are not real,
which is silly, annoying, and frustrating. But we’re pretty much stuck with it.

There are a couple of ways of describing the real numbers. I’m going to take you through a couple of them: first, an informal intuitive description; then an axiomatic definition, and finally, a constructive definition.

The Reals, Informally

The informal, intuitive description is the basic number line. Think about
a line, that goes on forever in both directions. There’s one spot on it labeled 0,
and on either side of zero it’s labeled like a ruler. Every point on that line is
a number. And given any two points on the line, there’s an infinite number of points
between them. The numbers that make up that line are the real numbers.


number-line.jpg

The intuition of the number line is a good one – most of the important properties
of the real numbers can be described in very nice, intuitive ways using the number
line. The ideas of addition, subtraction, ordering, and contintuity are all very
clear in the intuition of the number line; multiplication seems tricky, but that can be explained in terms of the number line (you can look at my posts about slide rules to get an idea of how.)

Assuming you’re not an insane math geek, you probably want to stop reading right here; I’m going to move on to more formal definitions of the reals, which most people probably don’t want to bother with. ūüôā

The Reals, Axiomatically

The axiomatic definition is, in many ways, quite similar to the definition
of the reals in terms of the number line – it just does it in a very formal way. An
Axiomatic definition doesn’t tell you how to get the real numbers – it just
describes them in terms of a set of rules in terms of simple set theory and logic.

The reals are defined by a tuple: (R,+,0,×,1,≤), where R is an infinite set; “+” and × are binary operators on members of R;
“0” and “1” are special distinguished elements of R; and ≤ is a binary
relation over members of R. The elements of the tuple must satisfy the following
axioms:

  • (R,+,×) are a field. What this means is:
    1. “+” and “×” are closed, total, and onto in R.
    2. “+” and “×” are commutative: a+b=b+a, a×b=b×a.
    3. “×” is distributive with respect to each “+”: (3+4)×5 = 3×5 + 4×5.
    4. 0 is the only identity value for “+”: For all a, a+0=a.
    5. For every member i∈R, there is exactly one
      value -i. called the additive inverse of i, so that
      i+-i=0, and for all i≠0, i≠-i.
    6. 1 is the only identity value for “×”; for all a, a×1=a.
    7. For every member i∈R except 0, there is
      exactly one
      value i-1, called the multiplicative inverse of i, such that i×i-1=1. For all i≠1, i≠i-1.
  • (R,≤) is a total order:
    1. For all members a,b∈ R, a≤b or b≤a.
    2. “≤” is transitive: if a≤b and b≤c then a≤c.
    3. “≤” is antisymmetric: if a≤b, and a≠b, then ¬(b≤a)
  • “≤” is compatible with “+” and “×”:
    1. If i≤j then (i+1)≤(j+1)
    2. If i≤j, then ∀x≥0, (i×x)≤(j×x)
    3. If i≤j, then ∀x≤0, (j×x)≤(i×x)
  • For every subset S⊂R where S≠∅, if S has an upper bound, then
    it has a least upper bound l such that for any x∈R that
    is an upper bound for S, l≤x.

That’s an extremely concise version of the axiomatic definition of reals. It describes what properties the real numbers must have, in terms of statements
that could be written out in first-order predicate logic form. An actual set of values that match that description is called a model for the definition;
you can show that there are models that match the definition, and that
all of the models that match the definition are equivalent.

The Reals, Constructively

Finally, the constructive definition. A constructive definitions is a
procedure for creating the set of real numbers.

We’ll start with the set of natural numbers. All of the natural numbers
are real numbers, with exactly the properties that they had as natural numbers.

Then we’ll add rational numbers. A positive rational number is defined by a pair of non-zero natural numbers called a ratio. A ratio n/d represents a real number which, when multiplied by “d”, gives the value “n”. The set of ratios numbers constructed this way ends up with lots of duplicates – 1/2, 2/4, 3/6, etc; so we ‘ll define the rationals as a set of equivalence classes over the ratios. To define ratio equality, we need to have multiplicative inverses: given a ratio a/b, the its inverse (a/b)-1 is b/a. Using this, we can say that two ratios n/d and m/e are equivalent if n/d × (m/e)-1 = 1. Using this, we can also see that every natural number except zero has a multiplicative inverse – N-1= 1/N, because N×1/N = 1.

So each of the equivalence classes of the rationals is a real number. Now we’ve got the positive rationals. Next, we add the additive inverses of all of the reals we’ve accumulated so far: for every positive real number N, there is exactly one negative real number -N, such that N+-N=0.

That gives us the complete set of rationals. For convenience, we’ll use Q to represent the set of rational numbers. Now we’re kind of stuck. We know that there are irrational numbers – we can define them axiomatically, and they fit the axiomatic definition of reals. So we need to be able to construct them. But how?

There are a bunch of tricks. The one I’m going to use is based on something called Dedekind cuts. Dedekind cuts basically says that you can define a real number r as a pair (A,B) of sets: A is the set of rational numbers smaller than r; B is the set of real numbers larger than r. Because of the nature of the rationals, these two sets have really peculiar properties. The set A is a set
containing values smaller than some number r; but there is no largest value or A. B is similar – there is no smallest value in B. r
is the number in the gap between the two sets in the cut.

How does that get us the irrational numbers? Here’s a simple example: let’s define
the square root of two using a Dedekind cut:

  • A = { r : r×r < 2 or r < 0}
  • B = { r : r×r > 2 and r > 0}

So we can say that the set of real numbers is: the set of numbers that can be defined using Dedekind cuts of the rationals.

We know that addition, multiplication, and comparisons work nicely on the rationals – they form a field, and they are totally ordered. Just to give you a sense of how we can show that the cuts also fit into that, we can show the definitions of “=”, “+”, and “≤” in terms of cuts.

Addition
The sum X+Y of two cuts X=(XL,XR) and Y=(YL,YR) = (ZL,ZR) where ZR = { x+y | x ∈ XR and y ∈ YR}.
Equality
Two cuts X=(XL,XR) and Y=(YL,YR) are equal if and only if XL ⊆ YL and XR ⊆ YR.
Ordering
Given two cuts X=(XL,XR) and Y=(YL,YR), X ≤ Y if and only if XL ⊆ YL and YR ⊆ XR.

0 thoughts on “Basics: Real Numbers

  1. oxeador

    I understand that your axiomatic definition is correct, and that it may be clearer this way, but your set of axioms is redundant. In the set of axioms that describe that R is a field, We do not need to request that + or x are onto. We do not need to request that 0 is the only identity for +, or that 1 is the only identity for x. We do not need to request that additive or multiplicative inverses are unique.

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  2. John Armstrong

    I’ll raise the same point I did for the natural numbers: Dedekind cuts don’t construct “the real numbers”. Those are defined by the axioms you gave (or a subset of those axioms, as oxeador points out).
    What Dedekind cuts construct is a model of the real numbers. That is, they show that if you have something satisfying the axioms of the rational numbers and a sufficient set-theoretical background you can construct something satisfying the axioms of the real numbers. Cauchy sequences construct a different model, but the two are structure-isomorphic. In fact, it can be shown that any two models of the real numbers are isomorphic, thus justifying the use of “the real numbers”.

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  3. Enigman

    What are the real numbers? They are the second-best of two worlds. Paradoxes (such as Burali-Forti’s) indicate that they are not a perfect model of actual continua (click on my ‘meaning’-full pseudonym for more such indications), whereas scientists actually work with finite subsets of the rational numbers (fuzzy ones at that). But the real numbers are algebraically simple, and closed enough for most purposes; and they are close enough to our intuitions about geometrical lines for most scientists to automatically assume a real number line when working with lines. They are a convenient stepping-stone, on our scientific way towards the truth (much as classical physics was).

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  4. Mark C. Chu-Carroll

    Blake:
    Yes, it is. In fact, the parallel between the Dedekind cuts and the Surreals is exactly what made me chose the cuts for the constructive definition. The surreals are a number system that’s practically designed to make CS geeks fall in love with ’em.
    I really need to revisit the surreals; I wrote a bit about them in the article you linked, but there’s so much more to say about them; and I think my presentation was a bit sloppy.

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  5. Nat Whilk

    John Armstrong’s claims are interesting to consider in the light of history. Dedekind worked out his theory of cuts in 1858, long before Hilbert developed his axiomatization of the reals in 1899, Zermelo developed his axiomatization of set theory in 1908, or model theory was invented in the mid 1900s. Thus, ostensibly, Dedekind created objects that weren’t real numbers but that with sufficient use of set theory would turn out to be merely a model of the real numbers decades before the real numbers were defined and set/model theory was invented. It’s this sort of pedantry that gives foundations a bad name among so many mathematicians.

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  6. Blake Stacey

    jimdesdu:
    Briefly put, irrational numbers are those real numbers which cannot be written as a ratio — get it, “ir-rational”? — of two integers. The square root of two is a good example. Transcendental numbers cannot be written as the solution of any polynomial equation with integer coefficients and a finite number of terms. All transcendental numbers are irrational, but not all irrational numbers are transcendental. The square root of two, for example, is clearly a solution to the polynomial equation x2 – 2 = 0.
    Transcendental numbers exist in infinite profusion. Familiar examples include π and e.

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  7. Harald Hanche-Olsen

    I kind of dislike Dedekind cuts because they make it hard to define the product of real numbers. You have to do it differently depending on the signs of the factors.
    My favourite construction is more algebraic: The real numbers is the ring of Cauchy sequences of rational numbers, modulo the maximal ideal therein consisting of the null sequences (which are sequences convering to zero).
    But I appreciate that this definition may be less accessible to beginners.

    Reply
  8. Mark C. Chu-Carroll

    Irrational numbers and transcendental numbers are a bit tricky – it depends a bit on how you choose to define them. It’s also a planned subject for another basics post.
    One thing you can say for certain: the transcendental numbers are a strict subset of the irrational numbers. But you can define both the irrationals and the transcendentals as being strictly reals, or as complex numbers.
    Rationals are numbers that can be described as ratios – fractions. That definition can be expanded to the complex numbers: complex rationals are complex numbers that can be described using only rational elements.
    Irrationals are numbers that cannot be described as ratios – numbers like π, or the square root of 2. Similarly, complex irrationals are numbers that cannot be expressed using only ratios.
    Transcendental numbers are irrational numbers that cannot be specified as the solution to an algebraic equation. So, for example, you can describe the square root of 2 in terms of algebra; it’s the solution to the equation x2=2. But you cannot describe π that way. The definition of transcendentals is the same in the world of complex numbers as it is for real numbers; numbers that cannot be described as the solution to an algebraic equation.

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  9. Jonathan Vos Post

    “Real” numbers were not called that until “imaginary numbers” were an accepted phrase. I agree with you, the words “real” and “imaginary” drag an unwanted ontological sense into the terminology. Electricval Engineers, in particular, consider “imaginary numbers” to have shocking reality.
    Consider the use of the term “hyperimaginary.” Are quaternions even less real than “imaginary” numbers? Are Octonions yet less real than quaternions?
    In retrospect, an unfortunate confusion in nomenclature. But the law of small numbers applies. There are only so many short words; pairs of them are pigeonholed into conflating distinct concepts.
    Vasiliev coined the term “imaginary logic” by analogy with “imaginary geometry” (Riemann), and in so doing provided important foundational work in metalogic and modal logic.

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  10. Blake Stacey

    Harald Hanche-Olsen:

    I kind of dislike Dedekind cuts because they make it hard to define the product of real numbers. You have to do it differently depending on the signs of the factors.

    Conway’s On Numbers and Games makes this point. He suggests constructing the integers and rationals in the familiar manner (basically as MarkCC has done here) and then building the reals as a subset of the surreals.

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  11. Coin

    Dedekind cuts basically says that you can define a real number r as a pair (A,B) of sets: A is the set of rational numbers smaller than r; B is the set of real numbers larger than r.
    Should the bolded word here be “rational”?
    If not, why not?

    Reply
  12. Cale Gibbard

    Enigman: The reals are important because they’re the unique complete ordered field. If you use anything else, then some of those properties are going to break. Completeness is important for analysis.
    Also, the Burali-Forti paradox has (basically) nothing to do with real numbers. It’s about the nonexistence of a set of all ordinal numbers. (Assuming one exists leads to a contradiction, hence the use of the term “paradox”.) I don’t really like the word “paradox”. There are no paradoxes, only differing consequences of varying sets of assumptions.
    Now, you might say that certain set theoretic axioms (such as the axiom of choice) allow one to find sets of reals with properties that you don’t like, but this is really a matter of taste, not at all one of truth.
    You might also say that the reals are not a good model for some of their applications in physics. If you want to make that claim and be taken seriously, you’d best find a better system, and show that it leads to better experimental predictions.

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  13. Blake Stacey

    Cale Gibbard wrote:

    You might also say that the reals are not a good model for some of their applications in physics. If you want to make that claim and be taken seriously, you’d best find a better system, and show that it leads to better experimental predictions.

    I keep waiting for the day when surreals turn out to be the proper tool for solving string theory/artificial intelligence/getting the shrink wrap off CDs.

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  14. Jon L

    “Before I go into detail, I need to say up front that I hate the term real number. It implies that other kinds of numbers are not real, which is silly, annoying, and frustrating.”
    This might be a random comment, but I too hated the phrase ‘the real world’ (referring to being part of the workforce, as opposed to being financially dependent on someone else) for pretty much the same reason. Reality is subjective.

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  15. Anonymous

    in terms of statements that could be written out in first-order predicate logic form
    The least upper bound condition is second-order. If all those axioms were first order, they would not uniquely characterize the reals, as Lowenheim-Skolem would be quick to point out. In fact, there would be a countable model of them.

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  16. Enigman

    Cale: I try to be interestingly controversial, and I end up being irritatingly condescending, sorry (I blame the nerdifying effect of using emails; what is it with emails?), and my point wasn’t helped by my saying “Burali-Forti” when I meant to say “Banach-Tarski” (what is it with similar foreign names?)… Still, “properties that [I] don’t like”? It’s more a matter of what properties real numbers really have; if not, then surely they are Not a perfect model of continua (assuming that there are physical continua, such as time, and prima facie there are).
    But anyway, what are the reals? I find it interesting that few here have regarded them as those numbers with endless decimal expansions (possibly ending with lots of zeroes, or repetitions). Isn’t that what most of us first take “real number” to be referring to? (Of course, they can be binary expansions too, as they tend to be in foundations, or in computing, or any other sort of expansion.) I mention this because one could argue that that way of thinking of the reals is best: not only is it more intuitive, it shows in a simple way how the nature of infinity creeps into the nature of the real numbers. If we assume the actual infinities of set theory (the transfinites) we get the standard real numbers; and with a potential infinity we would have that patternless real numbers (those that cannot have their endless expansions specified in a finite way) are not individual objects, but something more fuzzy, more like the product of an endless and incompletable process (e.g. imagine zooming in on a point, nesting it within narrower and narrower bounds, as we get an endless sequence of decimal places).

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  17. Miguel

    There is a very amusing constructive description. Namely, let’s say that the function Z -> Z is almost homomorphic if it’s boundary (co- ?) f(x+y)-f(x)-f(y) is bounded. For example, all affine functions f(x)=ax+b, a,b in Z are almost homomorphic. Clearly if f itself is bounded, it is almost homomorphic.
    The quotient of the group of almost homomorphic functions (w.r.t. the pointwise addition) by the subgroup of bounded functions is isomorphic to the additive group of reals.

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  18. Torbjörn Larsson

    In retrospect, an unfortunate confusion in nomenclature.

    It shows the contingency of history – square roots of negative numbers seemed ‘unreal’ at first.
    Similar contingencies is the choice of quadrant ordering in the cartesian system. And isn’t that the reason why the principal rotation around an axis is anti-clockwise?
    But if you think that is unfortunate, consider that in physics historical contingency made the charge of the ubiquitous electron negative. So in most materials (ie metals) the particle flow is anti-parallel to the charge flow, purely by convention.
    It is nice to have terms pointing out that numbers in the complex plane doesn’t behave like numbers in the cartesian plane, but I agree that the result could have been better.

    Transcendental numbers cannot be written as the solution of any polynomial equation with integer coefficients and a finite number of terms.

    Speaking of terms, I prefer to think of transcendental numbers as equivalently not expressible by polynomial equations with rational coefficients. That way I can think of them as transcending our powers to express them by rational numbers.

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  19. Torbjörn Larsson

    I meant to say “Banach-Tarski”

    But doesn’t that merely point to that using the axiom of choice gives some bizarre consequences?
    In any case, it doesn’t seem unique or peculiar that introducing idealizations or similar analytical tools gives some bizarre consequences. For example introducing infinities to simplify descriptions of unbounded quantities gives bizarre consequences when doing arithmetic operations.

    solving string theory/artificial intelligence/getting the shrink wrap off CDs.

    I see that you ordered that nicely according to increasing difficulty. Also, we may need a sufficiently powerful AI for that last task. ūüôā

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  20. Torbjörn Larsson

    But doesn’t that merely point to that using the axiom of choice gives some bizarre consequences?

    And, if according to above Dedekind cuts doesn’t use AC to give reals while Cauchy-sequences does (to claim an isomorphic model), it doesn’t seem fair to accuse the reals for your choice of using AC.

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  21. Thony C.

    “I’ll raise the same point I did for the natural numbers: Dedekind cuts don’t construct “the real numbers”. Those are defined by the axioms you gave (or a subset of those axioms, as oxeador points out)”
    Axiom systems cannot define mathematical objects. All they define is the rules for operating on those objects. For a detailed analysis as to why this is so, read the Frege-Hilbert correspondence where Frege makes this fundamental fact very clear.

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  22. BenE

    I have the opposite opinion as you Mark. I think real is a very appropriate nomenclature for the real numbers. The reason for this is simple. They are the numbers that are the closest to approximate the measures found in reality. At the limit, they represent best the continuous world we live in.
    I read Chaitin’s Meta Math book, and he like you seems to dislike much the concept of the reals. He argues at length how the reals are nothing but. The only argument that carries some weight in my opinion is the one from quantum physics which could in theory support a universe that is not continuous. But even there, given all the different interpretations of quantum phenomenon I think it is a very weak argument.
    All his other arguments only show that mathematics don’t do a good job of representing continuity since they depend on a number system that is grounded in discrete and distinct symbols or well fractioned and geometries.
    But all he shows are limits that seem to be entirely owned by mathematics, not by the real world. I think it makes sense to say that as you add decimals to a real number it tends towards representing a real dimension. The more you want precision the more you need digits which in practice limits this system of representation. However, this does not mean that it’s the world that isn’t continuous only that the representation of the world in mathematical language can only tend toward perfection when pushed at its limit.
    I think Chaitin is victim of what Jaynes calls the Mind Projection Fallacy which is when we falsely attribute to the real world a characteristic that should instead be attributed to our model or our method of representation. It is a flaw which Jaynes often attributes to frequencist probability theory.
    To Chaitin’s credit, he mentions (although very briefly) this view in his book (beginning of chapter 6 p120 in my edition) which he says is the opinion of Karl Svozil and maybe Einstein. But I’m guessing Svozil took it from Jaynes:
    “The physicist Karl Svozil has the following interesting position on these questions. Svozil has classical, deterministic leanings and sympathizes with Einstein’s assertion [(like Jayes)] that “God doesn’t play dice” Svozil admits that in its current state quantum theory contains randomness. But he thinks that this is only temporary, and that some new, deeper, hidden-variable theory will eventually restore determinacy and law to physics. On the other hand, Svozil believes that, as he puts it, ‘Omega’ shows that there is real randomness in the (unreal) mental mindscape fantasy world of pure math”
    I like him, think that math as a representation language created by man carries this randomness. But math is only a mental tool for us to approximate the real world. And real numbers are the best approximations we have to the real continuous world and thus deserve their name even if they are limited.
    Read more here:

    Reply
  23. Torbjörn Larsson

    The only argument that carries some weight in my opinion is the one from quantum physics which could in theory support a universe that is not continuous.

    I don’t think main stream physicists believe that. Because of lorentz covariance in the theories you can’t discretize spacetime. And without covariance you can’t have locality from local realism, so no individual objects.
    I agree that this is a bad argument, and that it looks like a projection. But you do it too here. Both when you discuss correspondences for frequentist probability and for reals.
    Since QM is our most fundamental description of reality, and QM is based on complex numbers, it is clear that complex numbers most closely correspond to it. By your standard, also frequentist probability corresponds best since QM is using it.
    Frequentists math is checked by being used successfully in fundamental and applied physics. If physicists are realists, and most are, it is their prerogative as scientists to define realism. Neither math nor philosophy can do that alone, because they are removed from observations. Though computer science seems close enough to help constrain physics, such as putting requirements on the anthropic principle.
    I am sympathetic to the idea of math as model. But the model is applied on an objective reality.

    But he thinks that this is only temporary, and that some new, deeper, hidden-variable theory will eventually restore determinacy and law to physics.

    Again, not a main stream physics thought. The Bell tests have nearly removed any little possibility for hidden variable theories.
    I think it is useful to separate between genuine randomness, which comes from QM interpretations (even many-worlds, since we can’t access the deterministic outcome), and randomness from coarsegraining.

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  24. √ėrjan Johansen

    I don’t think main stream physicists believe that. Because of lorentz covariance in the theories you can’t discretize spacetime. And without covariance you can’t have locality from local realism, so no individual objects.

    Loop quantum gravity seems to me to be getting more and more mainstream, probably because it shows promise of achieving exactly such an impossibility.

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  25. Torbjörn Larsson

    Loop quantum gravity seems to me to be getting more and more mainstream,

    I am not sure how to exactly measure this as a layman, but AFAIK for every LQG researcher there goes 10-20 string physicists which are much more productive on average.
    I’m not so hopeful for LQG since they can’t solve the simple harmonic oscillator or bound their energies from below. Perhaps it is a symptom of the localization problem.
    Not that string theory, which obeys lorentz covariance, hasn’t its own problems. ūüôā But these two basic hurdles aren’t among them. Instead, the problem is too many solutions and too few experiments.

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  26. Nathanael Nerode

    I tend to prefer the “Cauchy sequence” method of defining reals in terms of rationals.
    Why? Because it’s a lot closer than Dedekind cuts to the “natural” definition in terms of infinite series. (The natural series definition generates some annoying complexity in the exact definitions and theorems which is avoided by the sequence definition; I’ve gone through it, and you don’t gain much insight from it that you don’t get from the Cauchy sequence definition. The equivalency is worth proving though.)
    And because you have to prove a bunch of theorems about Cauchy sequences of real numbers in analysis anyway — since the theorems are very nearly the same as the ones needed to define the real numbers and prove their characteristics, it’s sort of a time saver.

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  27. Jesse

    The problem with the Cauchy sequence definition is that you need to be minimally comfortable with abstract algebra. At the very least you need to be to the point where you understand that if you have a ring R and a maximal ideal M then A/M is a field.
    Dedekind cuts, however, can be understood by people who have had a rigorous introduction to calculus. This is the approrach Spivak’s Calculus takes — the last two chapters involve constructing the reals from the rationals and proving that all totally ordered fields with the LUB property are field-isomorphic.

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