- Cardinality of the continuum
In

mathematics , the**cardinality of the continuum**, sometimes also called the**power of the continuum**, is the size (cardinality ) of the set ofreal numbers $mathbb\; R$ (sometimes called the continuum). The cardinality of $mathbb\; R$ is often denoted by $mathfrak\; c$. So, by definition, thecardinal number $mathfrak\; c\; =\; |mathbb\; R|.$Georg Cantor showed that the cardinality of the continuum is larger than that of the set ofnatural numbers $mathbb\{N\}$, namely $\{mathfrak\; c\}\; =\; 2^\{aleph\_0\},$ where $aleph\_0$ (aleph-null ) denotes the cardinality of $mathbb\{N\}$. In other words, although $mathbb\; R$ and $mathbb\; N$ are bothinfinite set s, the real numbers are in some sense "more numerous" than the natural numbers.**Intuitive argument**Every real number has an infinite

decimal expansion . For example, :^{1}/_{2}= 0.50000...:^{1}/_{3}= 0.33333...:$pi$ = 3.14159...Note that this is true even when 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 aone-to-one correspondence with the set of natural numbers $mathbb\{N\}$. This fact makes it sensible to talk about (for example) the first, the one-hundredth, or the millionth digit of $pi$. Since the natural numbers have cardinality $aleph\_0,$ each real number has $aleph\_0$ digits in its expansion. This is true no matter what mathematical base we are using, so for simplicity, let us consider a binary real number. Each position in its decimal expansion may hold either a 0 or a 1, so the number of all possible ways to fill those positions must be $2^\{aleph\_0\}.$ Therefore, the number of real numbers is $\{mathfrak\; c\}\; =\; 2^\{aleph\_0\}.$**Properties****Uncountability**Georg Cantor introduced the concept ofcardinality to compare the sizes of infinite sets. He famously showed that the set of real numbers isuncountably infinite ; i.e. $\{mathfrak\; c\}$ is strictly greater than the cardinality of thenatural numbers , $aleph\_0$::$aleph\_0\; <\; mathfrak\; c$In other words, there are strictly more real numbers than there are integers. Cantor proved this statement in several different ways. SeeCantor's first uncountability proof andCantor's diagonal argument .**Cardinal equalities**A variation on 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 itspower set , i.e. |"A"| < 2^{|"A"|}. One concludes that the power set "P"(**N**) of thenatural number s**N**is uncountable. It is then natural to ask whether the cardinality of "P"(**N**) is equal to $\{mathfrak\; c\}$. It turns out that the answer is yes. One can prove this in two steps:

#Define a map "f" :**R**→ "P"(**Q**) from the reals to the power set of therationals by sending each real number "x" to the set $\{q\; in\; mathbb\{Q\}\; mid\; q\; le\; x\}$ of all rationals less than or equal to "x" (with the reals viewed asDedekind cut s, this is nothing other than theinclusion map in the set of sets of rationals). This map isinjective since the rationals are dense in**R**. Since the rationals are countable we have that $mathfrak\; c\; le\; 2^\{aleph\_0\}$.

#Let {0,2}^{N}be the set of infinitesequence s with values in set {0,2}. This set clearly has cardinality $2^\{aleph\_0\}$ (the naturalbijection between the set of binary sequences and "P"(**N**) is given by theindicator function ). Now associate to each such sequence ("a"_{"i"}) the unique real number in the interval [0,1] with theternary -expansion given by the digits ("a"_{"i"}), i.e. the "i"-th digit after the decimal point is "a"_{"i"}. The image of this map is called theCantor set . It is not hard to see that this map isinjective , 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 $2^\{aleph\_0\}\; le\; mathfrak\; c$.By theCantor–Bernstein–Schroeder theorem we conclude that:$mathfrak\; c\; =\; |mathcal\{P\}(mathbb\{N\})|\; =\; 2^\{aleph\_0\}.$The cardinal equality $mathfrak\{c\}^2\; =\; mathfrak\{c\}$ can be demonstrated using

cardinal arithmetic : :$mathfrak\{c\}^2\; =\; (2^\{aleph\_0\})^2\; =\; 2^\{2\; imes\{aleph\_0\; =\; 2^\{aleph\_0\}\; =\; mathfrak\{c\}.$This argument is a condensed version of the notion of interleaving two binary sequences: let 0."a"_{0}"a"_{1}"a"_{2}… be the binary expansion of "x" and let 0."b"_{0}"b"_{1}"b"_{2}… be the binary expansion of "y". Then "z" = 0."a"_{0}"b"_{0}"a"_{1}"b"_{1}"a"_{2}"b"_{2}…, the interleaving of the binary expansions, is a well-defined function when "x" and "y" have unique binary expansions. Only countably many reals have non-unique binary expansions.By using the rules of cardinal arithmetic one can also show that:$mathfrak\; c^\{aleph\_0\}\; =\; \{aleph\_0\}^\{aleph\_0\}\; =\; n^\{aleph\_0\}\; =\; mathfrak\; c^n\; =\; aleph\_0\; mathfrak\; c\; =\; n\; mathfrak\; c\; =\; mathfrak\; c,$where "n" is any finite cardinal ≥ 2, and:$mathfrak\; c\; ^\{mathfrak\; c\}\; =\; (2^\{aleph\_0\})^\{mathfrak\; c\}\; =\; 2^\{mathfrak\; c\; imesaleph\_0\}\; =\; 2^\{mathfrak\; c\},$where $mathfrak\; c\; ^\{mathfrak\; c\}$ is the cardinality of the power set of

**R**, and $mathfrak\; c\; ^\{mathfrak\; c\}\; >\; mathfrak\; c$.**Beth numbers**The sequence of

beth number s is defined by setting $eth\_0\; =\; aleph\_0$ and $eth\_\{k+1\}\; =\; 2^\{eth\_k\}$. So $\{mathfrak\; c\}$ is the second beth number,**beth-one**::$mathfrak\; c\; =\; eth\_1$The third beth number,**beth-two**, is the cardinality of the power set of**R**(i.e. the set of all subsets of thereal line )::$2^mathfrak\; c\; =\; eth\_2.$**The continuum hypothesis**The famous

continuum hypothesis asserts that $\{mathfrak\; c\}$ is also the secondaleph number $aleph\_1$. In other words, the continuum hypothesis states that there is no set "A" whose cardinality lies strictly between $aleph\_0$ and $\{mathfrak\; c\}$:$otexists\; A\; :\; aleph\_0\; <\; |A|\; <\; mathfrak\; c$However, this statement is now known to be independent of the axioms ofZermelo-Fraenkel set theory (ZFC). That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzeronatural number "n", the equality $\{mathfrak\; c\}$ = $aleph\_n$ is independent of ZFC. (The case $n=1$ is 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 ofcofinality , e.g., $mathfrak\{c\}\; eqaleph\_omega.$ In particular, $mathfrak\{c\}$ could be either $aleph\_1$ or $aleph\_\{omega\_1\}$, where $omega\_1$ is the first uncountable ordinal, so it could be either asuccessor cardinal or alimit cardinal , and either aregular cardinal or asingular cardinal .

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

*the

real number s**R**

*any (nondegenerate) closed or open interval in**R**(such as theunit interval [0,1] )

*theirrational number s

*thetranscendental numbers

*Euclidean space **R**^{"n"}

*thecomplex number s**C**

*thepower set of thenatural number s (the set of all subsets of the natural numbers)

*the set ofsequences of integers (i.e. all functions**N**→**Z**, often denoted**Z**^{N})

*the set of sequences of real numbers,**R**^{N}

*the set of all continuous functions from**R**to**R**

*theCantor set

*theEuclidean topology on**R**^{"n"}(i.e. the set of allopen set s in**R**^{"n"})

*the Borel σ-algebra on**R**(i.e. the set of allBorel set s in**R**)

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

*the set of all subsets of

**R**, i.e., the power set of**R**, written "P"(**R**) or 2^{R}

*the set**R**^{R}of all functions from**R**to**R*** the Lebesgue σ-algebra of

**R**, i.e., the set of allLebesgue measurable sets in**R**.They all have cardinality $2^mathfrak\; c\; =\; eth\_2$ (ml|Beth number|Beth two|Beth two).

**References***

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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