Cardinality of the continuum

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In mathematics, the cardinality of the continuum (sometimes also called the power of the continuum) is the cardinal number of the set of real numbers \mathbb R (sometimes called the continuum). This cardinal number is often denoted by \mathfrak c, so \mathfrak c = |\mathbb R|.

Contents

[edit] Intuitive Argument

Every real number has an infinite decimal expansion. For example,

1/2 = 0.50000...
1/3 = 0.33333...
π = 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 a one-to-one correspondence with the set of natural numbers. This fact makes it sensible to talk about (for example) the first, the one-hundredth, or the millionth digit of π. Since the natural numbers have cardinality \aleph_0 (aleph-null), 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 {\mathfrak c} of all possible ways to fill those positions must be 2^{\aleph_0}. Therefore, {\mathfrak c}=2^{\aleph_0}.

[edit] Properties

[edit] 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; i.e. {\mathfrak c} is strictly greater than the cardinality of the natural 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. See Cantor's first uncountability proof and Cantor's diagonal argument.

[edit] 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 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 {\mathfrak c}. It turns out that the answer is yes. One can prove this in two steps:

  1. 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 \in \mathbb{Q} \mid q \le x\} 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). This map is injective since the rationals are dense in R. Since the rationals are countable we have that \mathfrak c \le 2^{\aleph_0}.
  2. Let {0,2}N be the set of infinite 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 (ai) the unique real number in the interval [0,1] with the ternary-expansion given by the digits (ai), i.e. the i-th digit after the decimal point is ai. 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 \mathfrak c.

By the Cantor–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\times{\aleph_0}} = 2^{\aleph_0} = \mathfrak{c}.

This argument is a condensed version of the notion of interleaving two binary sequences: let 0.a0a1a2… be the binary expansion of x and let 0.b0b1b2… be the binary expansion of y. Then z = 0.a0b0a1b1a2b2…, 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\times\aleph_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 .

[edit] Beth numbers

Main article: Beth number

The sequence of beth numbers is defined by setting \beth_0 = \aleph_0 and \beth_{k+1} = 2^{\beth_k}. So {\mathfrak c} is the second beth number, beth-one:

\mathfrak c = \beth_1

The third beth number, beth-two, is the cardinality of the power set of R (i.e. the set of all subsets of the real line):

2^\mathfrak c = \beth_2.

[edit] The continuum hypothesis

Main article: Continuum hypothesis

The famous continuum hypothesis asserts that {\mathfrak c} is also the second aleph 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}

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

However, this statement is now known to be independent of the axioms of Zermelo-Fraenkel set theory (ZFC). That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero natural 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 of cofinality, e.g., \mathfrak{c}\neq\aleph_\omega. In particular, \mathfrak{c} could be either \aleph_1 or \aleph_{\omega_1}, where ω1 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.

[edit] 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:

[edit] 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 2R
  • the set RR of all functions from R to R

Both have cardinality 2^\mathfrak c = \beth_2 (Beth twohttp://en.wikipedia.org../../../../articles/b/e/t/Beth_number.html#Beth_two).

[edit] 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.

This article incorporates material from cardinality of the continuum on PlanetMath, which is licensed under the GFDL.