Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers formula_1, sometimes called the continuum. It is an infinite cardinal number and is denoted by formula_2 (lowercase fraktur "c") or formula_3.
The real numbers formula_1 are more numerous than the natural numbers formula_5. Moreover, formula_1 has the same number of elements as the power set of formula_7 Symbolically, if the cardinality of formula_5 is denoted as formula_9, 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 formula_10 This is also true for several other infinite sets, such as any "n"-dimensional Euclidean space formula_11 (see space filling curve). That is,
The smallest infinite cardinal number is formula_9 (aleph-null). The second smallest is formula_13 (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between formula_9 and means that formula_15. The truth or falsity of this hypothesis is undecidable and cannot be proven within the widely used ZFC system of axioms.
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, formula_16 is strictly greater than the cardinality of the natural numbers, formula_9:
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, formula_18 (and so that the power set formula_19 of the natural numbers formula_5 is uncountable). In fact, one can show that the cardinality of formula_19 is equal to formula_16 as follows:
By the Cantor–Bernstein–Schroeder theorem we conclude that
The cardinal equality formula_42 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 formula_43 is the cardinality of the power set of R, and formula_44.
Alternative explanation for formula_45.
Every real number has at least one infinite decimal expansion. For example,
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 formula_46. 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 formula_47 each real number has formula_9 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 formula_49 to formula_50 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 formula_51 and formula_52. So formula_16 is the second beth number, beth-one:
The third beth number, beth-two, is the cardinality of the power set of formula_28 (i.e. the set of all subsets of the real line):
The continuum hypothesis.
The famous continuum hypothesis asserts that formula_16 is also the second aleph number, formula_13. In other words, the continuum hypothesis states that there is no set formula_57 whose cardinality lies strictly between formula_9 and formula_16
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 formula_16 = formula_61 is independent of ZFC (case formula_62 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., formula_63). In particular, formula_64 could be either formula_13 or formula_66, where formula_67 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 formula_16. Some common examples are the following:
Sets with greater cardinality.
Sets with cardinality greater than formula_16 include:
These all have cardinality formula_86 (beth two).