In set theory, the **cardinality of the continuum** is the cardinality or "size" of the set of real numbers , sometimes called the continuum. It is an infinite cardinal number and is denoted by (lowercase fraktur "c") or .^{ [1] }^{ [2] }

- Properties
- Uncountability
- Cardinal equalities
- Alternative explanation for c = 2 ℵ 0 {\displaystyle {\mathfrak {c}}=2^{\aleph {0}}}
- Beth numbers
- The continuum hypothesis
- Sets with cardinality of the continuum
- Sets with greater cardinality
- References
- Bibliography

The real numbers are more numerous than the natural numbers . 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 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 (*a*,*b*) is equinumerous with This is also true for several other infinite sets, such as any *n*-dimensional Euclidean space (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 .^{ [3] } The truth or falsity of this hypothesis is undecidable and cannot be proven within the widely used ZFC system of axioms.

Georg Cantor introduced the concept of cardinality 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 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.

A variation of Cantor's diagonal argument can be used to prove Cantor's theorem, 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 numbers is uncountable). In fact, one can show^{[ citation needed ]} that the cardinality of is equal to as follows:

- Define a map from the reals to the power set of the rationals, , by sending each real number to the set of all rationals less than or equal to (with the reals viewed as Dedekind cuts, this is nothing other than the inclusion map in the set of sets of rationals). Because the rationals are dense in , this map is injective, and because the rationals are countable, we have that .
- Let be the set of infinite sequences with values in set . This set has cardinality (the natural bijection between the set of binary sequences and is given by the indicator function). Now, associate to each such sequence the unique real number in the interval with the ternary-expansion given by the digits , i.e., , the -th digit after the fractional point is with respect to base . The image of this map is called the Cantor set. It is not hard to see that this map is injective, for by avoiding points with the digit 1 in their ternary expansion, we avoid conflicts created by the fact that the ternary-expansion of a real number is not unique. We then have that .

By the Cantor–Bernstein–Schroeder theorem 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 .

Every real number has at least one infinite decimal expansion. For example,

1/2 = 0.50000...

1/3 = 0.33333...

π = 3.14159....

(This is true even in the case the expansion repeats, as in the first two examples.)

In any given case, the number of digits is countable 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 digit 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

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):

The famous continuum hypothesis asserts that is also the second aleph number, .^{ [3] } 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 with the axiom of choice (ZFC). That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero natural number *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.

A great many sets studied in mathematics have cardinality equal to . Some common examples are the following:

- the real numbers
- any (nondegenerate) closed or open interval in (such as the unit interval )
- the irrational numbers
- the transcendental numbers We note that the set of real algebraic numbers is countably infinite (assign to each formula its Gödel number.) So the cardinality of the real algebraic numbers is . Furthermore, the real algebraic numbers and the real transcendental numbers are disjoint sets whose union is . Thus, since the cardinality of is , the cardinality of the real transcendental numbers is . A similar result follows for complex transcendental numbers, once we have proved that .
- the Cantor set
- Euclidean space
^{ [4] } - the complex numbers We note that, per Cantor's proof of the cardinality of Euclidean space,
^{ [4] }. By definition, any can be uniquely expressed as for some . We therefore define the bijection - the power set of the natural numbers (the set of all subsets of the natural numbers)
- the set of sequences of integers (i.e. all functions , often denoted )
- the set of sequences of real numbers,
- the set of all continuous functions from to
- the Euclidean topology on (i.e. the set of all open sets in )
- the Borel σ-algebra on (i.e. the set of all Borel sets in ).

Sets with cardinality greater than include:

- the set of all subsets of (i.e., power set )
- the set 2
^{R}of indicator functions defined on subsets of the reals (the set is isomorphic 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).

In mathematics, the **continuum hypothesis** is a hypothesis about the possible sizes of infinite sets. It states:

There is no set whose cardinality is strictly between that of the integers and the real numbers.

In mathematics, the **Cantor set** is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.

In mathematics, **cardinal numbers**, or **cardinals** for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The *transfinite* cardinal numbers, often denoted using the Hebrew symbol (aleph) followed by a subscript, describe the sizes of infinite sets.

In mathematics, the **cardinality** of a set is a measure of the "number of elements" of the set. For example, the set contains 3 elements, and therefore has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between the different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its **size**, when no confusion with other notions of size is possible.

In mathematics, especially in order theory, the **cofinality** cf(*A*) of a partially ordered set *A* is the least of the cardinalities of the cofinal subsets of *A*.

In mathematics, an **uncountable set** is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.

In set theory, **Cantor's diagonal argument**, also called the **diagonalisation argument**, the **diagonal slash argument**, the **anti-diagonal argument**, or the **diagonal method**, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.

In mathematics, **transfinite numbers** are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the **transfinite cardinals**, which are cardinal numbers used to quantify the size of infinite sets, and the **transfinite ordinals**, which are ordinal numbers used to provide an ordering of infinite sets. The term *transfinite* was coined by Georg Cantor in 1915, who wished to avoid some of the implications of the word *infinite* in connection with these objects, which were, nevertheless, not *finite*. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". Nevertheless, the term "transfinite" also remains in use.

In mathematics, particularly in set theory, the **aleph numbers** are a sequence of numbers used to represent the cardinality of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph.

In mathematics, two sets or classes *A* and *B* are **equinumerous** if there exists a one-to-one correspondence between them, that is, if there exists a function from *A* to *B* such that for every element *y* of *B*, there is exactly one element *x* of *A* with *f*(*x*) = *y*. Equinumerous sets are said to have the same cardinality. The study of cardinality is often called **equinumerosity** (*equalness-of-number*). The terms **equipollence** (*equalness-of-strength*) and **equipotence** (*equalness-of-power*) are sometimes used instead.

In mathematics, **limit cardinals** are certain cardinal numbers. A cardinal number *λ* is a **weak limit cardinal** if *λ* is neither a successor cardinal nor zero. This means that one cannot "reach" *λ* from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear.

In mathematics, the **beth numbers** are a certain sequence of infinite cardinal numbers, conventionally written , where is the second Hebrew letter (beth). The beth numbers are related to the aleph numbers, but there may be numbers indexed by that are not indexed by .

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a **function of a real variable** is a function whose domain is the real numbers ℝ, or a subset of ℝ that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the **real functions**, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.

In model theory, a branch of mathematical logic, the **spectrum of a theory** is given by the number of isomorphism classes of models in various cardinalities. More precisely, for any complete theory *T* in a language we write *I*(*T*, *α*) for the number of models of *T* of cardinality *α*. The **spectrum problem** is to describe the possible behaviors of *I*(*T*, *α*) as a function of *α*. It has been almost completely solved for the case of a countable theory *T*.

**Set theory of the real line** is an area of mathematics concerned with the application of set theory to aspects of the real numbers.

In set theory, **Cichoń's diagram** or **Cichon's diagram** is a table of 10 infinite cardinal numbers related to the set theory of the reals displaying the provable relations between these cardinal characteristics of the continuum. All these cardinals are greater than or equal to , the smallest uncountable cardinal, and they are bounded above by , the cardinality of the continuum. Four cardinals describe properties of the ideal of sets of measure zero; four more describe the corresponding properties of the ideal of meager sets.

In mathematics, a **cardinal function** is a function that returns cardinal numbers.

In mathematics, a **real number** is a value of a continuous quantity that can represent a distance along a line. The adjective *real* in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol **R** or and is sometimes called "the reals".

In the mathematical field of set theory, the **continuum** means the real numbers, or the corresponding (infinite) cardinal number, denoted by . Georg Cantor proved that the cardinality is larger than the smallest infinity, namely, . He also proved that is equal to , the cardinality of the power set of the natural numbers.

In the mathematical discipline of set theory, a **cardinal characteristic of the continuum** is an infinite cardinal number that may consistently lie strictly between , and the cardinality of the continuum, that is, the cardinality of the set of all real numbers. The latter cardinal is denoted or . A variety of such cardinal characteristics arise naturally, and much work has been done in determining what relations between them are provable, and constructing models of set theory for various consistent configurations of them.

- ↑ "Comprehensive List of Set Theory Symbols".
*Math Vault*. 2020-04-11. Retrieved 2020-08-12. - ↑ "Transfinite number | mathematics".
*Encyclopedia Britannica*. Retrieved 2020-08-12. - 1 2 Weisstein, Eric W. "Continuum".
*mathworld.wolfram.com*. Retrieved 2020-08-12. - 1 2 Was Cantor Surprised?, Fernando Q. Gouvêa,
*American Mathematical Monthly*, March 2011.

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*Naive set theory*. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). - Jech, Thomas, 2003.
*Set Theory: The Third Millennium Edition, Revised and Expanded*. Springer. ISBN 3-540-44085-2. - Kunen, Kenneth, 1980.
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*This article incorporates material from cardinality of the continuum on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*

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