, 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.
In other words, the open interval (a,b) is equinumerous with
, as well as with several other infinite sets, such as any n-dimensional Euclidean space
The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between
[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).
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.
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.
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.
[4][5] In one direction, reals can be equated with Dedekind cuts, sets of rational numbers,[4] or with their binary expansions.
[5] In the other direction, the binary expansions of numbers in the half-open interval
, 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.
Every real number has at least one infinite decimal expansion.
In any given case, the number of decimal places is countable since they can be put into a one-to-one correspondence with the set of natural numbers
This makes it sensible to talk about, say, the first, the one-hundredth, or the millionth decimal place of π.
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
and consider that decimal fractions containing only 3 or 7 are only a part of the real numbers, then we get and thus The sequence of beth numbers is defined by setting
(i.e. the set of all subsets of the real line): The continuum hypothesis asserts that
[2] In other words, the continuum hypothesis states that there is no set
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.
In fact, for every nonzero natural number n, the equality
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.
A great many sets studied in mathematics have cardinality equal to
Some common examples are the following: Per Cantor's proof of the cardinality of Euclidean space,[9]
We therefore define the bijection Sets with cardinality greater than
(beth two) This article incorporates material from cardinality of the continuum on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.