In set theory, the cardinality of the continuum is the cardinality or "size" of the Set (mathematics), set of real numbers $\backslash mathbb\; R$, sometimes called the Continuum (set theory), continuum. It is an Infinite set, infinite cardinal number and is denoted by $\backslash mathfrak\; c$ (lowercase fraktur "c") or $,\; \backslash mathbb\; R,$.
The real numbers $\backslash mathbb\; R$ are more numerous than the natural numbers $\backslash mathbb\; N$. Moreover, $\backslash mathbb\; R$ has the same number of elements as the power set of $\backslash mathbb\; N.$ Symbolically, if the cardinality of $\backslash mathbb\; N$ is denoted as aleph number#Aleph-nought, $\backslash aleph\_0$, the cardinality of the continuum is
This was proven by Georg Cantor in his Cantor's first uncountability proof, uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his Cantor's diagonal argument, 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 $\backslash mathbb\; R.$ This is also true for several other infinite sets, such as any ''n''-dimensional Euclidean space $\backslash mathbb\; R^n$ (see space filling curve). That is,
The smallest infinite cardinal number is $\backslash aleph\_0$ (aleph number#Aleph-nought, aleph-null). The second smallest is $\backslash aleph\_1$ (aleph number#Aleph-one, aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between $\backslash aleph\_0$ and means that $\backslash mathfrak\; c\; =\; \backslash aleph\_1$. The truth or falsity of this hypothesis is undecidable and Continuum_hypothesis#Independence_from_ZFC, cannot be proven within the widely used ZFC system of axioms.

^{R} of indicator functions defined on subsets of the reals (the set $2^$ is isomorphic to $\backslash mathcal(\backslash mathbb)$ – the indicator function chooses elements of each subset to include)
*the set $\backslash mathbb^\backslash mathbb$ of all functions from $\backslash mathbb$ to $\backslash mathbb$
*the Lebesgue measure, Lebesgue σ-algebra of $\backslash mathbb$, i.e., the set of all Lebesgue measurable sets in $\backslash mathbb$.
*the set of all Lebesgue integration, Lebesgue-integrable functions from $\backslash mathbb$ to $\backslash mathbb$
*the set of all Measurable function, Lebesgue-measurable functions from $\backslash mathbb$ to $\backslash mathbb$
*the Stone–Čech compactifications of $\backslash mathbb$, $\backslash mathbb$ and $\backslash mathbb$
*the set of all automorphisms of the (discrete) field of complex numbers.
These all have cardinality $2^\backslash mathfrak\; c\; =\; \backslash beth\_2$ (Beth number#Beth two, beth two).

Properties

Uncountability

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, $\backslash aleph\_0$: 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, see Cantor's first uncountability proof and Cantor's diagonal argument.Cardinal equalities

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, $,\; A,\; <\; 2^$ (and so that the power set $\backslash wp(\backslash mathbb\; N)$ of the natural numbers $\backslash mathbb\; N$ is uncountable). In fact, one can show that the cardinality of $\backslash wp(\backslash mathbb\; N)$ is equal to $$ as follows: #Define a map $f:\backslash mathbb\; R\backslash to\backslash wp(\backslash mathbb\; Q)$ from the reals to the power set of the rationals, $\backslash mathbb\; Q$, by sending each real number $x$ to the set $\backslash $ of all rationals less than or equal to $x$ (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 set, dense in $\backslash mathbb$, this map is Injective function, injective, and because the rationals are countable, we have that $\backslash mathfrak\; c\; \backslash le\; 2^$. #Let $\backslash ^$ be the set of infinite sequences with values in set $\backslash $. This set has cardinality $2^$ (the natural bijection between the set of binary sequences and $\backslash wp(\backslash mathbb\; N)$ is given by the indicator function). Now, associate to each such sequence $(a\_i)\_$ the unique real number in the unit interval, interval $[0,1]$ with the Ternary numeral system, ternary-expansion given by the digits $a\_1,a\_2,\backslash dotsc$, i.e., $\backslash sum\_^\backslash infty\; a\_i3^$, the $i$-th digit after the fractional point is $a\_i$ with respect to base $3$. 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 $2^\; \backslash le\; \backslash mathfrak\; c$. By the Cantor–Bernstein–Schroeder theorem we conclude that The cardinal equality $\backslash mathfrak^2\; =\; \backslash mathfrak$ 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 $2\; ^$ is the cardinality of the power set of R, and $2\; ^\; >\; \backslash mathfrak\; c$.Alternative explanation for $=\; 2^$

Every real number has at least one infinite decimal 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 digits is countable set, countable since they can be put into a one-to-one correspondence with the set of natural numbers $\backslash mathbb$. 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 $\backslash aleph\_0,$ each real number has $\backslash aleph\_0$ 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 $2\; =\; \backslash $ to $\backslash $ and consider that decimal fractions containing only 3 or 7 are only a part of the real numbers, then we get and thusBeth numbers

The sequence of beth numbers is defined by setting $\backslash beth\_0\; =\; \backslash aleph\_0$ and $\backslash beth\_\; =\; 2^$. So $$ is the second beth number, beth-one: The third beth number, beth-two, is the cardinality of the power set of $\backslash mathbb$ (i.e. the set of all subsets of the real line):The continuum hypothesis

The famous continuum hypothesis asserts that $$ is also the second aleph number, $\backslash aleph\_1$. In other words, the continuum hypothesis states that there is no set $A$ whose cardinality lies strictly between $\backslash aleph\_0$ and $$ 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 $$ = $\backslash aleph\_n$ is independent of ZFC (case $n=1$ 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 (set theory), König's theorem on the grounds of cofinality (e.g., $\backslash mathfrak\backslash neq\backslash aleph\_\backslash omega$). In particular, $\backslash mathfrak$ could be either $\backslash aleph\_1$ or $\backslash aleph\_$, where $\backslash omega\_1$ 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 $\backslash mathbb$ (i.e., power set $\backslash mathcal(\backslash mathbb)$) *the set Power set#Representing subsets as functions, 2References

Bibliography

*Paul Halmos, ''Naive set theory''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition). *Thomas Jech, Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. . *Kenneth Kunen, 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