Completeness Axiom and the Real Numbers
Lunga
02 August 2023
What do we really mean by Completeness?
We won’t provide a specific definition but will approach it in a way that
makes intuitive sense.
The real numbers can be defined synthetically as an ordered field satisfying
some version of the completeness axiom. Different versions of this axiom are
all equivalent in the sense that any ordered field that satisfies one form of com-
pleteness satisfies all of them, apart from Cauchy completeness and the nested
intervals theorem, which are strictly weaker in that there are non-Archimedean
fields that are ordered and Cauchy complete. When the real numbers are in-
stead constructed using a model, completeness becomes a theorem or collection
of theorems.
OK, let’s break it down. If you watch the lectures that I suggested
for real analysis, you are free to skip this section
Simple explanation
1. What Are Real Numbers?
Real numbers are a type of number we use in math. These numbers, you
will soon find out that they have (1) The least upper bound property and
(2) Are Dedekind complete.
2. Explanation
(a) What is an ordered field?:
Definition: An ordered set S, is a set with a relation
⊑
which satisfies the following properties:
i. ∀x, y ∈, S with x = y, x > y or x < y ( Any 2 or more elements
in that set are comparable)
ii. if x > y and y > z then x > z (Transitivity)
1
, (b) Archimedian Property: This property is vital for understanding
the Completeness Axiom. Basically this property tells us that the
Natural numbers (N) are unbounded in R. In the following session,
Lunga (Me) will briefly explain what do we mean by being ”bounded”
and ”convergent” in two different but complementary ways.
Proof: (*)If x ∈ R, y ∈ R, and x > 0, then there is a positive
integer n such that nx > y.
Let A be the set of all nx, where n runs through the posi-
tive integers. If (*) were false, then y would be an upper
bound of A. But then A has a least upper bound in R. Put
α = sup A. Since x > 0, α − x < α, and α − x is not an upper
bound of A. Hence α − x < mx for some positive integer m.
But then α < (m + 1)x ∈ A, which is impossible, since α is an
upper bound of A.
Why is (b) important?: This question is key to understanding Analysis of
the real number set. Well, without the Archimedian property, the completeness
axioms somewhat lose their power, they are strictly weaker than they are in the
real number set. Sets/ Fields, where the Archimedian property does not hold
are referred to as being non-Archimedian fields. The completeness axioms in
Non-Archimedian, ordered sets and Cauchy complete are less ”effective” than
they are in R. Hence, in R, completeness becomes a theorem that encapsulates
a collection of theorems as stated above.
Conclusion: Completeness is a fundamental property that characterizes the
real numbers and distinguishes them from other number systems. Whether it
is taken as an axiom or proven as a theorem, completeness ensures that there
are no ”gaps” or missing points on the real number line, making it an essential
concept in real analysis.
1 Forms of Completeness
1.1 Least Upper Bound Property
The least upper bound property states that any set of real numbers with an
upper limit has a smallest upper limit within the set. This ensures that there
are no ”gaps” above a set of numbers. For example, in the rational numbers,
not every set has a least upper bound.
Explanation of LUB: The least-upper-bound property is a way of making
sure that the real numbers are complete, meaning they don’t have any missing
2
Lunga
02 August 2023
What do we really mean by Completeness?
We won’t provide a specific definition but will approach it in a way that
makes intuitive sense.
The real numbers can be defined synthetically as an ordered field satisfying
some version of the completeness axiom. Different versions of this axiom are
all equivalent in the sense that any ordered field that satisfies one form of com-
pleteness satisfies all of them, apart from Cauchy completeness and the nested
intervals theorem, which are strictly weaker in that there are non-Archimedean
fields that are ordered and Cauchy complete. When the real numbers are in-
stead constructed using a model, completeness becomes a theorem or collection
of theorems.
OK, let’s break it down. If you watch the lectures that I suggested
for real analysis, you are free to skip this section
Simple explanation
1. What Are Real Numbers?
Real numbers are a type of number we use in math. These numbers, you
will soon find out that they have (1) The least upper bound property and
(2) Are Dedekind complete.
2. Explanation
(a) What is an ordered field?:
Definition: An ordered set S, is a set with a relation
⊑
which satisfies the following properties:
i. ∀x, y ∈, S with x = y, x > y or x < y ( Any 2 or more elements
in that set are comparable)
ii. if x > y and y > z then x > z (Transitivity)
1
, (b) Archimedian Property: This property is vital for understanding
the Completeness Axiom. Basically this property tells us that the
Natural numbers (N) are unbounded in R. In the following session,
Lunga (Me) will briefly explain what do we mean by being ”bounded”
and ”convergent” in two different but complementary ways.
Proof: (*)If x ∈ R, y ∈ R, and x > 0, then there is a positive
integer n such that nx > y.
Let A be the set of all nx, where n runs through the posi-
tive integers. If (*) were false, then y would be an upper
bound of A. But then A has a least upper bound in R. Put
α = sup A. Since x > 0, α − x < α, and α − x is not an upper
bound of A. Hence α − x < mx for some positive integer m.
But then α < (m + 1)x ∈ A, which is impossible, since α is an
upper bound of A.
Why is (b) important?: This question is key to understanding Analysis of
the real number set. Well, without the Archimedian property, the completeness
axioms somewhat lose their power, they are strictly weaker than they are in the
real number set. Sets/ Fields, where the Archimedian property does not hold
are referred to as being non-Archimedian fields. The completeness axioms in
Non-Archimedian, ordered sets and Cauchy complete are less ”effective” than
they are in R. Hence, in R, completeness becomes a theorem that encapsulates
a collection of theorems as stated above.
Conclusion: Completeness is a fundamental property that characterizes the
real numbers and distinguishes them from other number systems. Whether it
is taken as an axiom or proven as a theorem, completeness ensures that there
are no ”gaps” or missing points on the real number line, making it an essential
concept in real analysis.
1 Forms of Completeness
1.1 Least Upper Bound Property
The least upper bound property states that any set of real numbers with an
upper limit has a smallest upper limit within the set. This ensures that there
are no ”gaps” above a set of numbers. For example, in the rational numbers,
not every set has a least upper bound.
Explanation of LUB: The least-upper-bound property is a way of making
sure that the real numbers are complete, meaning they don’t have any missing
2
