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Construction Of The Real Number System
Gary Sng Chee Hien and Denny Leung Ho-Hon1
Department of Mathematics, National University of Singapore
2 Science Drive 2, Singapore 117543.
ABSTRACT
Currently, there is no modules in the mathematics department of NUS that offer a
rigorous treatment of the construction of the real number system. The real number
system is simply taken for granted, with all its associated properties. Although this
current approach of assuming the existence of the real number system is unarguably
reasonable and practical for most undergraduate mathematics, it does have several
drawbacks.
Firstly, since the real number system is so fundamental to mathematics, at least at the
undergraduate level, the omission of the treatment of its construction would seem to
represent a knowledge gap in the average mathematics student’s training. Secondly,
although most mathematics student may causally dismiss the real number system as
intuitively obvious, there is cause for concern that this perspective was borne out of
familiarity, and not intuition. For example, can anyone easily visualize an irrational
number or give an intuitive argument for its existence? Why is it that the order
completeness property holds in the real number system, or , from an axiomatic
perspective, why does mathematicians ascribe to the real number system the order
completeness property? When confronted with such questions, one may find that the
real number system may not be so intuitive after all, even though it is such a familiar
system.
On an intuitive level, the real number is certainly deeper than the natural numbers,
integers or rationals. We can give physical analogy to the above 3 types of numbers
easily, but can we also give a physical analogy to an irrational number? All of these
sums up to one conclusion. The real number system is not intuitive at all. We do not
know where it came from. Neither do we know the reason for its existence. Yet we
are taught the mechanics of the real number system and have become so familiar with
it that we never ever question its existence! For all we know, the real number system
may turn out to be a ‘false’ system and this would make most of the mathematics that
we learnt meaningless. There is hence a real need to question the existence of the real
number system.
The aim of this project is to construct the real number system by starting with the
axioms of Peano, which basically assume a counting system that modeled the natural
number system. Most of the concepts employed in the construction are actually
concepts taught at the undergraduate level, thereby demonstrating that it is viable to
perhaps set up an undergraduate module to expound such a construction. Also,
concepts such as integral domain, ring, field etc are actually borne out through a deep
understanding of the various number systems, and so exposure to a rigorous
construction of the real number system would enable the student to better appreciate
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Associate Professor
the motivation behind such concepts. We will briefly outline the approach used in the
construction and mention some of the associated concepts.
The Natural Number System
We start by assuming the axioms of Peano, which gives us the rules to create a
natural number system:
The five axioms of Peano
We assume the existence of a set ℕ with the following properties:
N(i) There exists an element 1ℕ
N(ii) For every nℕ, there exists an element S(n) ℕ such that
{(n,S(n))| n ℕ} is a function.
N(iii) 1 S(ℕ)
N(iv) S is one-one.
N(v) If P is any subset of ℕ such that 1P and S(n)P ∀ nP, then
P=ℕ.
The construction proceed to show that such a natural number system is unique and
that we can define the familiar addition, multiplication and exponentiation on it. Order
is also defined and we then proceed to show that the system is well-ordered, order
complete and that it possess the Archimedean Property.
The Integer System
We start by creating an integer system from the natural number system by means of
defining an equivalence relation on the set of pairs of natural numbers. We then
formally gives the definition for a system to be an integer system:
The integer system
An ordered integral domain (X, , ⊙,≻) is called an integer system if  a subset ℕX
of X such that
(i) Both (ℕX, ) and (ℕX, ⊙) are semi-groups and under the same isomorphism
:ℕXℕ, we have (ℕX, )≃ (ℕ,+) and (ℕX, ⊙)≃ (ℕ, .) as semi-group.
Furthermore, for every x, y ℕX, we have x≻y  (x)> (y)
(ii) For every xX,  y, zℕX s.t x = y  (-z)
The important conclusion here is that integer system exists and that it is also unique.
We demonstrate several important concepts associated with integers and show that
they are relevant also within our technical framework. In particular, we proved the
fundamental theorem of Arithmetic. We also showed that the integer system is order
complete but not well-ordered.
The Field Of Rational
A field of rationals is similarly also created from the integer system by means of
defining a suitable equivalence relation. We then formally gives the definition for a
system to be a field of rational:
Field of Rational
An ordered field (F, ⊕, ⊙,≻) is called a field of rationals if  an ordered subdomain
(Fℤ, ⊕, ⊙,≻) such that
(i) (Fℤ, ⊕, ⊙,≻) ≃ (ℤ, +, . , >)
(ii) For every xF,  y, zFℤ s.t x = y-1⊙z
It is similarly shown to be an ordered field and that it is unique. Familiar results such
as the denseness of rationals are also proved. This portion of the construction is
significant, as we will argue intuitively that the rationals are inadequate and that an
extension of the rational system is necessary. This is basically demonstrated with the
drawing of a triangle with a side that is non-rational in length. The technical statement
can be loosely represented by the theorem that say that the rational system is not order
complete. We will also prove the seemingly unrelated fact that the rational system is
not Cauchy complete, but the construction of the actual real number system will
reveal that this two concepts are in fact inexorably linked.
The Real Number System
For the construction of the actual real number system, we actually gives two such
constructions. One was by using Dedekind’s cut and the other an approach by Georg
Cantor using Cauchy sequences. Both started off with seemingly different assumption
of what the real number system should be. Dedekind’s cut was used to extend the
field of rational to an order complete field while Cantor’s approach aims to make it
Cauchy complete instead. Their formal approach is given below:
The Dedekind’s Cut
A subset, , of ℚ is called a cut if it has the following properties:
ℝ(i)   ,   ℚ
ℝ(ii) For every r and sℚ\, r < s
ℝ(iii) Max does not exist.
The Dedekind Real Number System
An ordered field (ℝD, ⊕,⊙, ≻) is called a Dedekind real number system if
(i)
There exist a subfield (ℚD, ⊕,⊙, ≻) which is isomorphic to (ℚ,+,.,>).
(ii)
(ℝD, ⊕,⊙, ≻) is order complete.
Cantor’s Approach
Let C denote the set of all Cauchy sequences in ℚ. We say (rn), (sn) C are
equivalent and we write (rn) ∼ (sn) if given any rational  > 0,  kℕ such that
| rn – sn | < 
 n  k.
The Cantor Real Number System
For any ordered field (ℝC, ⊕, ⊙, ≻), we say that it is a Cantor Real Number System
if
(i)  a subfield (ℚC, ⊕, ⊙, ≻) that is isomorphic to (ℚ, +, ., >).
(ii) (ℝC, ⊕, ⊙, ≻) is Cauchy Complete
(iii) (ℝC, ⊕, ⊙, ≻) has the Archimedean property.
By following two approaches, we do not contradict our philosophy that we do not
know what the real number system should be. We simply found two incompleteness
of the rational system, and we proceed to complete each one separately to see what
we obtain. The wonderful result is that both Dedekind and Cantor’s approach ended
up with a common number system, and that this system is unique. We will then argue
that this common number system is suitable to be the real number system, and is in
fact the real number system that we have been using all along.
Conclusion.
The main aim of the project is to demonstrate that the existence of the real number
system is not that simple and obvious but a rigorous verification on the part of the
student is not impossible. We choose to start with the axioms of Peano since that, in
the author’s opinion, spelt out a natural system that can be intuitively accepted. At the
very least, it is more intuitively clear than the axiom that the real number system must
be order complete! Also, the construction process makes use only of fundamental
concepts which is accessible to the normal mathematics undergraduate, thereby
demonstrating that diverse branches of mathematics and seemingly unrelated concepts
can actually be pieced together to weave a beautiful system.
REFERENCES
Inder K.Rana(1998), From Numbers To Analysis, River Edge, NJ : World Scientific.
Robert G.Bartle, Donald R. Sherbert(1992), Introduction To Real Analysis 2nd
Edition, John Wiley and Sons, Inc.