In
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, the cardinality of the continuum is the
cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
or "size" of the
set of
real numbers
In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
, sometimes called the
continuum. It is an
infinite cardinal number
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
and is denoted by
(lowercase
Fraktur "c") or
The real numbers
are more numerous than the
natural numbers
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
. Moreover,
has the same number of elements as the
power set of
. Symbolically, if the cardinality of
is denoted as
, the cardinality of the continuum is
This was proven by
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
in his
uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his
diagonal argument in 1891. Cantor defined cardinality in terms of
bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them.
Between any two real numbers ''a'' < ''b'', no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers. In other words, the
open interval
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
(''a'',''b'') is
equinumerous with
, as well as with several other infinite sets, such as any ''n''-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
(see
space filling curve). That is,
The smallest infinite cardinal number is
(
aleph-null). The second smallest is
(
aleph-one). The
continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between
and , means that
.
The truth or falsity of this hypothesis is undecidable and
cannot be proven within the widely used
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
with axiom of choice (ZFC).
Properties
Uncountability
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
introduced the concept of
cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
to compare the sizes of infinite sets. He famously showed that the set of real numbers is
uncountably infinite. That is,
is strictly greater than the cardinality of the
natural numbers
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
,
:
In practice, this means that there are strictly more real numbers than there are integers. Cantor proved this statement in several different ways. For more information on this topic, see
Cantor's first uncountability proof and
Cantor's diagonal argument
Cantor's diagonal argument (among various similar namesthe diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof) is a mathematical proof that there are infin ...
.
Cardinal equalities
A variation of Cantor's diagonal argument can be used to prove
Cantor's theorem
In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any Set (mathematics), set A, the set of all subsets of A, known as the power set of A, has a strictly greater cardinality than A itself.
For finite s ...
, which states that the cardinality of any set is strictly less than that of its
power set. That is,
(and so that the power set
of the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s
is uncountable). In fact, the cardinality of
, by definition
, is equal to
. This can be shown by providing one-to-one mappings in both directions between subsets of a countably infinite set and real numbers, and applying the
Cantor–Bernstein–Schroeder theorem according to which two sets with one-to-one mappings in both directions have the same cardinality.
In one direction, reals can be equated with
Dedekind cuts, sets of rational numbers,
[ or with their binary expansions.][ In the other direction, the binary expansions of numbers in the half-open interval , viewed as sets of positions where the expansion is one, almost give a one-to-one mapping from subsets of a countable set (the set of positions in the expansions) to real numbers, but it fails to be one-to-one for numbers with terminating binary expansions, which can also be represented by a non-terminating expansion that ends in a repeating sequence of 1s. This can be made into a one-to-one mapping by that adds one to the non-terminating repeating-1 expansions, mapping them into .][ Thus, we conclude that][
The cardinal equality can be demonstrated using cardinal arithmetic:
By using the rules of cardinal arithmetic, one can also show that
where ''n'' is any finite cardinal ≥ 2 and
where is the cardinality of the power set of R, and .
]
Alternative explanation for
Every real number has at least one infinite decimal expansion
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\cdots b_0.a_1a_2\cdots
Here is the decimal separator ...
. For example,
(This is true even in the case the expansion repeats, as in the first two examples.)
In any given case, the number of decimal places is countable
In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
since they can be put into a one-to-one correspondence with the set of natural numbers . This makes it sensible to talk about, say, the first, the one-hundredth, or the millionth decimal place of π. Since the natural numbers have cardinality each real number has digits in its expansion.
Since each real number can be broken into an integer part and a decimal fraction, we get:
where we used the fact that
On the other hand, if we map to and consider that decimal fractions containing only 3 or 7 are only a part of the real numbers, then we get
and thus
Beth numbers
The sequence of beth numbers is defined by setting and . So is the second beth number, beth-one:
The third beth number, beth-two, is the cardinality of the power set of (i.e. the set of all subsets of the real line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
):
The continuum hypothesis
The continuum hypothesis asserts that is also the second aleph number, . In other words, the continuum hypothesis states that there is no set whose cardinality lies strictly between and
This statement is now known to be independent of the axioms of Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
with the axiom of choice (ZFC), as shown by Kurt Gödel and Paul Cohen. That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
''n'', the equality = is independent of ZFC (case being the continuum hypothesis). The same is true for most other alephs, although in some cases, equality can be ruled out by König's theorem on the grounds of cofinality (e.g. ). In particular, could be either or , where is the first uncountable ordinal, so it could be either a successor cardinal or a limit cardinal, and either a regular cardinal or a singular cardinal.
Sets with cardinality of the continuum
A great many sets studied in mathematics have cardinality equal to . Some common examples are the following:
Sets with greater cardinality
Sets with cardinality greater than include:
*the set of all subsets of (i.e., power set )
*the set 2R of indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
s defined on subsets of the reals (the set is isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to – the indicator function chooses elements of each subset to include)
*the set of all functions from to
*the Lebesgue σ-algebra of , i.e., the set of all Lebesgue measurable sets in .
*the set of all Lebesgue-integrable functions from to
*the set of all Lebesgue-measurable functions from to
*the Stone–Čech compactifications of , , and
*the set of all automorphisms of the (discrete) field of complex numbers.
These all have cardinality ( beth two)
See also
* Cardinal characteristic of the continuum
References
Bibliography
*Paul Halmos
Paul Richard Halmos (; 3 March 1916 – 2 October 2006) was a Kingdom of Hungary, Hungarian-born United States, American mathematician and probabilist who made fundamental advances in the areas of mathematical logic, probability theory, operat ...
, ''Naive set theory''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition).
* Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. .
* Kunen, Kenneth, 1980. '' Set Theory: An Introduction to Independence Proofs''. Elsevier. .
{{PlanetMath attribution, urlname=CardinalityOfTheContinuum, title=cardinality of the continuum
Cardinal numbers
Set theory
Infinity