The system of real numbers is perhaps the most important mathematical invention. It grew out of the notion of a number as a quantity, as a measurement. It is used in every branch of mathematics, and the elucidation of its properties has occupied mathematics from the ancient Greeks to the present day.
The real numbers lie in a hierarchy of number systems, of increasing size:
We define the real numbers by three groups of axioms. These are:
A field is a common and much-studied algebraic system . Among the number systems listed above, the rational numbers, real numbers, and complex numbers are examples of fields. Both the rational numbers and the real numbers can be ordered in a line, in addition to the field properties. What is distinctive about the real numbers is completeness: any set of real numbers that is bounded above has a least upper bound.
Summary of the field axioms
Summary of the order axioms
The completeness axiom
Let `S` be a nonempty set of real numbers.
A number `z` is an upper bound of `S` if `x le z` for any `x in S`.
A number `y` is a least upper bound of `S` if `y` is an upper bound, and for any upper
bound `z`, we have `y le z`.
The completeness axiom says that any nonempty set of real numbers with an upper bound has a least upper bound.
Categoricity
The axioms for the real numbers are categorical. This means that for any two structures that satisfy the axioms, there is a unique isomorphism that preserves addition, multiplication, and order.There are a number of ways to construct the real numbers. We give here three standard constructions:
It is sometimes convenient to extend the real number system with two infinite values: `infty` and `-infty`. We will say that the ordinary real numbers are finite numbers and the two new numbers are infinite numbers, and call the enlarged system the extended real number system.
The completeness axiom states that any nonempty set `S` of real numbers that is bounded above has a least upper bound. That is, there is a real number `alpha` such that:
In an analogous way, every nonempty set `S` of real numbers that is bounded below has a greatest upper bound. This number is called `text(inf)(S) =` the infinum of `S`, or the greatest lower bound of `S`.
Some consequences of completeness:
Theorem: (The Archimedean Principle)
For every real number `x`, there is an integer `n` such that `n > x`.
Proof: Assume not; then the set of all integers `m le x` is a bounded set of real numbers, so it
has a least upper bound `N`. The integer `N+1` is therefore `> x`.
Theorem (Density of Rationals)
For any real numbers `x,y` such that `x < y`, there is a rational
number `r` such that `x < r < y`.
Proof: Let `q` be an integer ` > 1/(y-x)`. Then `qy - qx > 1`. The set of integers `ge qy` is bounded below,
so it has a greatest lower bound, an integer `p`. Then `qx < p-1 < qy`. The rational `r = (p-1)/q` is between `x` and `y`.
We can generalize slightly the notion of infinum and supremum to the extended real number system. Define `text(sup)(O/)=-infty` and `text(inf)(O/)=+infty`.
Theorem Every set of extended real numbers has a supremum (possibly `+-infty`) and an infinum (possibly `+-infty`).
A central concept of the real numbers is the idea of a limit, of a sequence of numbers that approach another number. We give the usual definition of a limit:
Definition:
Let `A = (. a_1, a_2, ... .)` be a sequence of real numbers, and let `L` be a real number. We define
The idea of a limit may be refined: we can define an upper limit and lower limit of a sequence of real numbers.
Definition: The upper limit of `(. a_1, a_2, ... .)`, written `text(lim sup)_(n->infty)a_n` , is defined to be
Theorem: A sequence of real numbers has a limit `L` iff it has both an upper limit of `L` and a lower limit `L`.
The cardinal of a set is the size of a set, the number of elements in a set. For finite sets, the cardinality
is the number of elements in the set. But the notion of cardinality extends to infinite sets.
In set theory, there are various constructions to define the cardinal of a set. >> Here we will use the concept informally.
We say that two sets `A, B` have the same cardinality (written `|A| = |B|`) if there is a bijection from `A` onto `B`.
We say the cardinality of `A` is less than or equal to the cardinality of `B` (written `|A| le |B|`) if
there is an injection from `A` into `B`.
It is evident that equality of cardinals is an equivalence relation among sets. Also, if `|A| le |B|` and `|B| le |C|`, then `|A| le |C|`. A central theorem in the theory of cardinals, called the Cantor-Bernstein theorem, says that if `|A| le |B|` and `|B| le |A|`, then `|A| = |B|`. (>)
For finite sets, the notion of the cardinal of a set is very intuitive. If A has three distinct elements, then `|A|=|B|` iff `B` has three elements, and `|A| le |B|` iff `B` has three or more elements. Cantor extended the theory to infinite sets.
A set `A` is countable if `|A| le |NN|`, where `NN` is the set of natural numbers = `{0, 1, 2, ...}`.
Theorem (Cantor) The set of real numbers is not countable.
So the set of real numbers cannot be exhausted by a sequence: a sequence is a smaller infinity than the infinity of the real numbers. We write `c` as the cardinal of the continuum.
An infinite subset of the real line may be at most countable, or it may be equivalent to `RR`, the entire set of reals. Is there another alternative? The Continuum Hypothesis is the statement that there is no other alternative.
The real line is a linearly ordered set, and has the following order properties:
Theorem: If `(L,<)` is a dense, unbounded, separable, complete linear order,
then `L` is order-isomorphic to the real line.
Sketch of a proof: Cantor's theorem on dense linear orders says that any two countable dense
unboundeed linear orders are order-isomorphic. This is proved by the so-called "back and forth" argument.
So if `L` and `L'` are linear orders that both satisfy the hypotheses, and `S`, `S'`, are countable
dense subsets, then let `f` be an order isomorphism from `S` to `S'`, and use the completeness properties
to extend `f` to all elements of `L` and `L'`.
The Suslin Problem
A closed interval in a linear order `L` is a set of the form `{x in L: a le x le b}`, where `a le b` are
elements of `L`. A closed interval is nontrivial if `a lt b`.
A linear order `L` satisfies the countable chain condition if every collection of pairwise disjoint
nontrivial closed intervals is countable.
Suslin raised the question on whether we could characterize the real line as a dense unbounded complete linear order that satisfies the countable chain condition. (Note that separability implies the countable chain condition.) We call a linear order a Suslin line if it is dense, unbounded, complete, not separable, and satisfies the countable chain condition. A Suslin line would give a negative answer to Suslin's question.
It turns out the existence of a Suslin line is independent of the usual axioms of set theory. To mention one result of an extensive theory: a Suslin line exists in `L`, the universe of constructible sets.
Some related linear orders
The long line. The Aronszajn line. The Surreal Numbers
The real line has a natural metric or distance functions `d`: for any two real numbers `x` and `y`, define the distance between them as `d(x,y) = | x - y |`. The real numbers with this metric form a metric space >> The standard metric on the real line is given by `d(x,y) = |x-y|`. This gives a topology on the set of real numbers.
Some properties of the standard metric
Complete, separable metric spaces
Much of the study of the real numbers in descriptive set theory centers on complete separable metric spaces
(also called Polish spaces). Two closely related spaces are Baire space and the Cantor space.
The Baire space consists of all
infinite sequences of non-negative integers, and can be metrized so that if two sequences that
agree on exactly the first `m` coordinates, then their distance is `2^(-m)`. The Cantor
space is the subspace of the Baire space consisting of sequences of 0's and 1's. The reals,
Baire space, and Cantor space are all examples of Polish spaces, of complete separable
metric spaces.
Other examples:
Every closed subspace of a Polish space is a Polish space.
Any product of a finite number of Polish spaces is a Polish space.
Theorem:
The standard metric defines a topology on the set of real numbers. The same topology can be defined as an order topology, with a basis of open intervals.
Some topological properties of the reals:
Other topologies.
The Sorgenfrey topology on the real numbers is the generated by the basis of all half-open intervals: `cc(B) = {[a,b) : a,b in RR, a lt b}`. The topology is distinct from the standard topology. For example, the real line under the Sorgenfrey topology is totally disconnected. Another topology, the open ray topology, has a base consisting of all open rays:
Topological characterization of the real line
First we observe that real line is homeomorphic to any open interval `(a,b)` and to any open ray `(a,infty)`. Similarly, any closed ray `[a,infty)` is homeomorphic to any half-closed interval `[a,b)`, and any nontrivial closed interval is homeomorphic to any other. But these three types of intervals - open, half-open, closed - are topologically distinct. (To see this, note that we can remove at most zero, one, or two points and still have a connected space.)
A point `x` in a connected space `X` is a cut point if `X setminus {x}` is not connected; otherwise it is a non-cut point. Clearly, every point in the real line is a cut point.
Theorem
If a space `X` is compact, connected, metrizable, and has exactly two non-cut points, then `X` is homeomorphic
to a closed interval.
If a space `X` is metrizable, separable, connected, locally connected, and has the property
that `X setminus {x}` has exactly two components for every `x in X`, then `X` is homeomorphic to the real line.
(A. J. Ward, 1936)
Intuitively, a `n`-dimensional (topological) manifold is a topological space in which every neighborhood is homeomorphic to Euclidean `n`-space. The interest is in the variety of global structures possible, given a Euclidean local structure.
Definition of a manifold.
Theorem: (Classification of 1-manifolds.)
The real numbers may be viewed as a line, a continuum without beginning or end. A future expansion of the text will discuss some geometric aspects of the real line.
A future expansion of the text will discuss computable real numbers and the classification of real numbers by Turing degrees.
Definition:
Let `K` be a field. A valuation is a function from `K` into the positive real numbers such that:Example: the trivial valuation. For any field `K`, define
A valuation defines a metric on a field: the distance between `x` and `y` is `phi(y-x)`.
`p-`adic valuations of the rationals.
Let `p` be a prime. We define the valuation `phi_p` as follows:
First, define `phi_p(0) = 0`. Next, every nonzero rational number `r` may be written in the form
`r = q/(p^(alpha)*s)` where `q,s` are integers not divisible by `p`.
Define `phi_p(r) = p^(alpha)`.
Theorem `phi_p` is a valuation.
Proof: Let `x,y,z in QQ`.
The p-Adic Real Numbers
The `p-`adic real numbers are the metric completion of the rational numbers using the valuation `phi_p`. A 2-adic real number may be thought of as an infinite binary sequence to the left of the decimal point, and a finite binary sequence to the right of the decimal point.Ostrowski's Theorem
The only valuation completions of the rational numbers are:Algebraic and transcendental numbers
A real number is algebraic iff it is a root of a polynomial with integer coefficients. Every rational number is algebraic, and some irrational numbers are algebraic. For example, `sqrt(2)` is a root of the polynomial `x^2-2`. and so is an algebraic number. A real number that is not algebraic is transcendental. These definitions extend to complex numbers. Theorem
The algebraic numbers form a subfield of the complex number field.
The real algebraic numbers form a subfield of the real number field.
Theorem
The set of all algebraic numbers is countable.
This implies that almost all numbers are transcendental, a result first proved by Cantor.
Let `P(x) = c_0 + c_1 x^1 + ldots + c_n x^n` be any polynomial with complex coefficients.
Define the height of `P`, writtem `H(P)`, to be
The Mahler classification says that `z` is
Theorem The `A`-numbers are precisely the algebraic numbers.
Theorem The `U_1` numbers are precisely the Liouville numbers.