Cardinality of the continuum

In

${\displaystyle \mathbb {R} }$, sometimes called the continuum. It is an infinite cardinal number and is denoted by ${\displaystyle {\mathbf {\mathfrak {c}}}}$ (lowercase Fraktur "c") or ${\displaystyle {\mathbf {|}}{\mathbf {\mathbb {R} }}{\mathbf {|}}}$^[1]

The real numbers ${\displaystyle \mathbb {R} }$ are more numerous than the

${\displaystyle \mathbb {N} }$. Moreover, ${\displaystyle \mathbb {R} }$ has the same number of elements as the power set of ${\displaystyle \mathbb {N} }$. Symbolically, if the cardinality of ${\displaystyle \mathbb {N} }$ is denoted as ${\displaystyle \aleph _{0}}$, the cardinality of the continuum is

This was proven by

: 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

with ${\displaystyle \mathbb {R} }$. This is also true for several other infinite sets, such as any n-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ (see ). That is,

The smallest infinite cardinal number is ${\displaystyle \aleph _{0}}$ (aleph-null). The second smallest is ${\displaystyle \aleph _{1}}$ (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between ${\displaystyle \aleph _{0}}$ and ${\displaystyle {\mathfrak {c}}}$, means that ${\displaystyle {\mathfrak {c}}=\aleph _{1}}$.^[2] The truth or falsity of this hypothesis is undecidable and cannot be proven within the widely used Zermelo–Fraenkel set theory with axiom of choice (ZFC).

Properties

Uncountability

. That is, ${\displaystyle {\mathfrak {c}}}$ is strictly greater than the cardinality of the , ${\displaystyle \aleph _{0}}$:

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

.

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, ${\displaystyle |A|<2^{|A|}}$ (and so that the power set ${\displaystyle \wp (\mathbb {N} )}$ of the natural numbers ${\displaystyle \mathbb {N} }$ is uncountable).^[3] In fact, the cardinality of ${\displaystyle \wp (\mathbb {N} )}$, by definition ${\displaystyle 2^{\aleph _{0}}}$, 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 the

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]

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

:

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

where n is any finite cardinal ≥ 2 and

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

Alternative explanation for 𝔠 = 2^א_‎0

Every real number has at least one infinite

. 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

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 ${\displaystyle \aleph _{0},}$ each real number has ${\displaystyle \aleph _{0}}$ 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 ${\displaystyle 2=\{0,1\}}$ 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

and thus

Beth numbers

The sequence of beth numbers is defined by setting ${\displaystyle \beth _{0}=\aleph _{0}}$ and ${\displaystyle \beth _{k+1}=2^{\beth _{k}}}$. So ${\displaystyle {\mathfrak {c}}}$ is the second beth number, beth-one:

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 the

):

The continuum hypothesis

The continuum hypothesis asserts that ${\displaystyle {\mathfrak {c}}}$ is also the second aleph number, ${\displaystyle \aleph _{1}}$.^[2] In other words, the continuum hypothesis states that there is no set ${\displaystyle A}$ whose cardinality lies strictly between ${\displaystyle \aleph _{0}}$ and ${\displaystyle {\mathfrak {c}}}$

This statement is now known to be independent of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), as shown by Kurt Gödel and Paul Cohen.^[6][7][8] That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero natural number n, the equality ${\displaystyle {\mathfrak {c}}}$ = ${\displaystyle \aleph _{n}}$ 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 by König's theorem on the grounds of cofinality (e.g. ${\displaystyle {\mathfrak {c}}\neq \aleph _{\omega }}$). In particular, ${\displaystyle {\mathfrak {c}}}$ could be either ${\displaystyle \aleph _{1}}$ or ${\displaystyle \aleph _{\omega _{1}}}$, where ${\displaystyle \omega _{1}}$ is the

.

Sets with cardinality of the continuum

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

Sets with greater cardinality

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 set 2^R of indicator functions defined on subsets of the reals (the set ${\displaystyle 2^{\mathbb {R} }}$ is
  isomorphic
  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 ${\displaystyle \mathbb {R} }$
• the Lebesgue σ-algebra of ${\displaystyle \mathbb {R} }$, i.e., the set of all
  Lebesgue measurable
  sets in ${\displaystyle \mathbb {R} }$.
• the set of all Lebesgue-integrable functions from ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} }$
• the set of all Lebesgue-measurable functions from ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} }$
• the Stone–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)

See also

• Cardinal characteristic of the continuum

References

Bibliography

• ISBN 0-387-90092-6
  (Springer-Verlag edition).
• ISBN 3-540-44085-2
  .
• ISBN 0-444-86839-9
  .

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