Inset theory, thecardinality of the continuum is thecardinality or "size" of theset ofreal numbers${\displaystyle \mathbb{R}}$, sometimes called thecontinuum. It is aninfinitecardinal number and is denoted by${\displaystyle \mathfrak{c}}$ (lowercaseFraktur "c") or${\displaystyle \mathbf{|}\mathbb{R}\mathbf{|}.}$^[1]

The real numbers${\displaystyle \mathbb{R}}$ are more numerous than thenatural numbers${\displaystyle \mathbb{N}}$. Moreover,${\displaystyle \mathbb{R}}$ has the same number of elements as thepower set of${\displaystyle \mathbb{N}}$. Symbolically, if the cardinality of${\displaystyle \mathbb{N}}$ is denoted as${\displaystyle {\mathrm{\aleph }}_0}$, the cardinality of the continuum is

${\displaystyle \mathfrak{c}=2^{{\mathrm{\aleph }}_0}>{\mathrm{\aleph }}_0.}$

This was proven byGeorg Cantor in hisuncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in hisdiagonal argument in 1891. Cantor defined cardinality in terms ofbijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them.

Between any two real numbersa < 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, theopen interval (a,b) isequinumerous with${\displaystyle \mathbb{R}}$, as well as with several other infinite sets, such as anyn-dimensionalEuclidean space${\displaystyle {\mathbb{R}}^n}$ (seespace filling curve). That is,

${\displaystyle |(a,b)|=|\mathbb{R}|=|{\mathbb{R}}^n|.}$

The smallest infinite cardinal number is${\displaystyle {\mathrm{\aleph }}_0}$ (aleph-null). The second smallest is${\displaystyle {\mathrm{\aleph }}_1}$ (aleph-one). Thecontinuum hypothesis, which asserts that there are no sets whose cardinality is strictly between${\displaystyle {\mathrm{\aleph }}_0}$ and${\displaystyle \mathfrak{c}}$, means that${\displaystyle \mathfrak{c}={\mathrm{\aleph }}_1}$.^[2] This hypothesis isindependent of the widely usedZermelo–Fraenkel set theory with axiom of choice (ZFC); that is, ZFC can neither prove that it is true nor that it is false.

Georg Cantor introduced the concept ofcardinality to compare the sizes of infinite sets. He famously showed that the set of real numbers isuncountably infinite. That is,${\displaystyle \mathfrak{c}}$ is strictly greater than the cardinality of thenatural numbers,${\displaystyle {\mathrm{\aleph }}_0}$:

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, seeCantor's first uncountability proof andCantor's diagonal argument.

A variation of Cantor's diagonal argument can be used to proveCantor's theorem, which states that the cardinality of any set is strictly less than that of itspower set. That is, (and so that the power set${\displaystyle \mathrm{\wp }(\mathbb{N})}$ of thenatural numbers${\displaystyle \mathbb{N}}$ is uncountable).^[3] In fact, the cardinality of${\displaystyle \mathrm{\wp }(\mathbb{N})}$, by definition${\displaystyle 2^{{\mathrm{\aleph }}_0}}$, is equal to${\displaystyle \mathfrak{c}}$. 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 theCantor–Bernstein–Schroeder theorem according to which two sets with one-to-one mappings in both directions have the same cardinality.^[4][5] In one direction, reals can be equated withDedekind cuts, sets of rational numbers,^[4] or with theirbinary expansions.^[5] In the other direction, the binary expansions of numbers in the half-open interval${\displaystyle [0,1)}$, 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${\displaystyle [1,2)}$.^[5] Thus, we conclude that^[4][5]

${\displaystyle \mathfrak{c}=|\mathrm{\wp }(\mathbb{N})|=2^{{\mathrm{\aleph }}_0}.}$

The cardinal equality${\displaystyle {\mathfrak{c}}^2=\mathfrak{c}}$ can be demonstrated usingcardinal arithmetic:

${\displaystyle {\mathfrak{c}}^2=(2^{{\mathrm{\aleph }}_0}{)}^2=2^{2\times {\mathrm{\aleph }}_0}=2^{{\mathrm{\aleph }}_0}=\mathfrak{c}.}$

By using the rules of cardinal arithmetic, one can also show that

${\displaystyle {\mathfrak{c}}^{{\mathrm{\aleph }}_0}={{\mathrm{\aleph }}_0}^{{\mathrm{\aleph }}_0}=n^{{\mathrm{\aleph }}_0}={\mathfrak{c}}^n={\mathrm{\aleph }}_0\mathfrak{c}=n\mathfrak{c}=\mathfrak{c}}$

wheren is any finite cardinal ≥ 2 and

${\displaystyle {\mathfrak{c}}^{\mathfrak{c}}=(2^{{\mathrm{\aleph }}_0}{)}^{\mathfrak{c}}=2^{\mathfrak{c}\times {\mathrm{\aleph }}_0}=2^{\mathfrak{c}}}$

where${\displaystyle 2^{\mathfrak{c}}}$ is the cardinality of the power set ofR, and${\displaystyle 2^{\mathfrak{c}}>\mathfrak{c}}$.

Every real number has at least one infinitedecimal expansion. 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 iscountable since they can be put into aone-to-one correspondence with the set of natural numbers${\displaystyle \mathbb{N}}$. 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${\displaystyle {\mathrm{\aleph }}_0,}$ each real number has${\displaystyle {\mathrm{\aleph }}_0}$ digits in its expansion.

Since each real number can be broken into an integer part and a decimal fraction, we get:

${\displaystyle \mathfrak{c}\le {\mathrm{\aleph }}_0\cdot {10}^{{\mathrm{\aleph }}_0}\le 2^{{\mathrm{\aleph }}_0}\cdot {(2^4)}^{{\mathrm{\aleph }}_0}=2^{{\mathrm{\aleph }}_0+4\cdot {\mathrm{\aleph }}_0}=2^{{\mathrm{\aleph }}_0}}$

where we used the fact that

${\displaystyle {\mathrm{\aleph }}_0+4\cdot {\mathrm{\aleph }}_0={\mathrm{\aleph }}_0}$

On the other hand, if we map${\displaystyle 2=\{0,1\}}$ to${\displaystyle \{3,7\}}$ and consider that decimal fractions containing only 3 or 7 are only a part of the real numbers, then we get

${\displaystyle 2^{{\mathrm{\aleph }}_0}\le \mathfrak{c}}$

and thus

${\displaystyle \mathfrak{c}=2^{{\mathrm{\aleph }}_0}.}$

The sequence of beth numbers is defined by setting${\displaystyle {\beth }_0={\mathrm{\aleph }}_0}$ and${\displaystyle {\beth }_{k+1}=2^{{\beth }_k}}$. So${\displaystyle \mathfrak{c}}$ is the second beth number,beth-one:

${\displaystyle \mathfrak{c}={\beth }_1.}$

The third beth number,beth-two, is the cardinality of the power set of${\displaystyle \mathbb{R}}$ (i.e. the set of all subsets of thereal line):

${\displaystyle 2^{\mathfrak{c}}={\beth }_2.}$

The continuum hypothesis asserts that${\displaystyle \mathfrak{c}}$ is also the secondaleph number,${\displaystyle {\mathrm{\aleph }}_1}$.^[2] In other words, the continuum hypothesis states that there is no set${\displaystyle A}$ whose cardinality lies strictly between${\displaystyle {\mathrm{\aleph }}_0}$ and${\displaystyle \mathfrak{c}}$

This statement is now known to be independent of the axioms ofZermelo–Fraenkel set theory with the axiom of choice (ZFC), as shown byKurt Gödel andPaul Cohen.^[6][7][8] That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzeronatural numbern, the equality${\displaystyle \mathfrak{c}}$ =${\displaystyle {\mathrm{\aleph }}_n}$ is independent of ZFC (case${\displaystyle n=1}$ being the continuum hypothesis). The same is true for most other alephs, although in some cases, equality can be ruled out byKönig's theorem on the grounds ofcofinality (e.g.${\displaystyle \mathfrak{c}\ne {\mathrm{\aleph }}_{\omega }}$). In particular,${\displaystyle \mathfrak{c}}$ could be either${\displaystyle {\mathrm{\aleph }}_1}$ or${\displaystyle {\mathrm{\aleph }}_{{\omega }_1}}$, where${\displaystyle {\omega }_1}$ is thefirst uncountable ordinal, so it could be either asuccessor cardinal or alimit cardinal, and either aregular cardinal or asingular cardinal.

A great many sets studied in mathematics have cardinality equal to${\displaystyle \mathfrak{c}}$. Some common examples are the following:

Sets with cardinality greater than${\displaystyle \mathfrak{c}}$ include:

• the set of all subsets of${\displaystyle \mathbb{R}}$ (i.e., power set${\displaystyle \mathcal{P}(\mathbb{R})}$)
• the set2^R ofindicator functions defined on subsets of the reals (the set${\displaystyle 2^{\mathbb{R}}}$ isisomorphic to${\displaystyle \mathcal{P}(\mathbb{R})}$ – the indicator function chooses elements of each subset to include)
• the set${\displaystyle {\mathbb{R}}^{\mathbb{R}}}$ of all functions from${\displaystyle \mathbb{R}}$ to${\displaystyle \mathbb{R}}$
• theLebesgue σ-algebra of${\displaystyle \mathbb{R}}$, i.e., the set of allLebesgue measurable sets in${\displaystyle \mathbb{R}}$.
• the set of allLebesgue-integrable functions from${\displaystyle \mathbb{R}}$ to${\displaystyle \mathbb{R}}$
• the set of allLebesgue-measurable functions from${\displaystyle \mathbb{R}}$ to${\displaystyle \mathbb{R}}$
• theStone–Čech compactifications of${\displaystyle \mathbb{N}}$,${\displaystyle \mathbb{Q}}$, and${\displaystyle \mathbb{R}}$
• the set of all automorphisms of the (discrete) field of complex numbers.

These all have cardinality${\displaystyle 2^{\mathfrak{c}}={\beth }_2}$ (beth two).

• Cardinal characteristic of the continuum

• Paul Halmos,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.Set Theory: An Introduction to Independence Proofs. Elsevier.ISBN 0-444-86839-9.

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