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Wikipedia / Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers, sometimes called the continuum. It is an infinite cardinal number (lowercase Fraktur "c

are more numerous than the natural numbers has the same number of elements as the power set is denoted as \aleph_0, 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, as well as with several other infinite sets, such as any n-dimensional Euclidean space (see space filling curve). That is,

(aleph-null (aleph-one). The continuum hypothesis and . This hypothesis is independent of the widely used Zermelo–Fraenkel set theory with axiom of choice (ZFC); that is, ZFC can neither prove that it is true nor that it is false.

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 is strictly greater than the cardinality of the natural numbers: 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 of the natural numbers is uncountable).. 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 Cantor–Bernstein–Schroeder theorem according to which two sets with one-to-one mappings in both directions have the same cardinality. In one direction, reals can be equated with Dedekind cuts, sets of rational numbers, or with their binary expansions.. Thus, we conclude that

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

is the cardinality of the power set of R.

Alternative explanation for &cfr; 2א‎0

Every real number has at least one infinite decimal expansion. 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 countable since they can be put into a one-to-one correspondence 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

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

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

The continuum hypothesis

is also the second aleph number. 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. That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero natural number n 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 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

. Some common examples are the following:

Sets with greater cardinality

include:

(beth two).

See also

References

Bibliography