cardinality of the continuum

In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). This cardinal number is often denoted by c,

c = |R|

Properties

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; i.e. c is strictly greater than the cardinality of the natural numbers, ℵ₀ (aleph-null)

\aleph_0 < c

In other words, there are strictly more real numbers than there are integers. Cantor proved this statement in a couple of different ways. See Cantor's first uncountability proof and Cantor's diagonal argument. 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 its power set, i.e. |A| < 2^|A|. One concludes that the power set P(N) of the natural numbers N is uncountable. It is then natural to ask whether the cardinality of P(N) is equal to 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 the rationals by sending each real number x to the set {q ∈ Q | q ≤ r} of all rationals less than or equal to x. This map is injective since the rationals are dense in R. Since the rationals are countable we have that $c \le 2^{\aleph_0}$ .
Let {0,2}^N be the set of sequences with values in set {0,2}. This set clearly has cardinality $2^{\aleph_0}$ (the natural bijection between the set of binary sequences and P(N) is given by the indicator function). Now associate to each such sequence (a_i) the unique real number in the inteval 0,1 with the ternary-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 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 $2^{\aleph_0} \le c$ .

By the Cantor-Bernstein-Schroeder theorem we conclude that

c = |P(\mathbb{N})| = 2^{\aleph_0}.

The sequence of beth numbers is defined by setting

\beth_0 = \aleph_0

and

\beth_{k+1} = 2^{\beth_k}

. So c is the first beth number, beth-one

c = \beth_1

The second beth number,

\beth_2 = 2^c

, is the set of all subsets of the real line. By using the rules of cardinal arithmetic one can show that

c = n c = \aleph_0 c = c^n = n^{\aleph_0} = {\aleph_0}^{\aleph_0} = c^{\aleph_0}

where n is any finite cardinal ≥ 2.

The continuum hypothesis

The famous continuum hypothesis asserts that c is also the first aleph number ℵ₁. In other words, the continuum hypothesis states that there is no set A whose cardinality lies strictly between ℵ₀ and c

\not\exists A : \aleph_0 < |A| < c

However, this statement is now known to be independent of the axioms of Zermelo-Fraenkel set theory. That is, both the hypothesis and its negation are consistent with these axioms.

Sets with cardinality c

A great many sets studied in mathematics have cardinality equal to c. Some common examples are the following:

the real numbers R
any (nondegenerate) closed or open interval in R (such as the unit interval 0,1)
the irrational numbers
the transcendental numbers
Euclidean space Rⁿ
the power set of the natural numbers (the set of all subsets of the natural numbers)
the set of sequences 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 (the set of all functions R^R has cardinality 2^c)
the Cantor set
the Euclidean topology on Rⁿ (i.e. the set of all open sets in Rⁿ)

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