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, 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. That is, (and so that the power set of the natural numbers is uncountable). In fact, one can show that the cardinality of is equal to as follows: #Define a map from the reals to the power set of the rationals, , by sending each real number to the set of all rationals less than or equal to (with the reals viewed as Dedekind cuts, this is nothing other than the inclusion map in the set of sets of rationals). Because the rationals are Dense set, dense in , this map is Injective function, injective, and because the rationals are countable, we have that . #Let be the set of infinite sequences with values in set . This set has cardinality (the natural bijection between the set of binary sequences and is given by the indicator function). Now, associate to each such sequence the unique real number in the unit interval, interval with the Ternary numeral system, ternary-expansion given by the digits , i.e., , the -th digit after the fractional point is with respect to base . 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 . By the Cantor–Bernstein–Schroeder theorem we conclude that The cardinal equality 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 is the cardinality of the power set of R, and .Alternative explanation for
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 digits is countable set, 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 digit of π. Since the natural numbers have cardinality each real number has 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 to and consider that decimal fractions containing only 3 or 7 are only a part of the real numbers, then we get and thusBeth numbers
The sequence of beth numbers is defined by setting and . So is the second beth number, beth-one: The third beth number, beth-two, is the cardinality of the power set of (i.e. the set of all subsets of the real line):The continuum hypothesis
The famous continuum hypothesis asserts that is also the second aleph number, . In other words, the continuum hypothesis states that there is no set whose cardinality lies strictly between and 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 = is independent of ZFC (case 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 (set theory), König's theorem on the grounds of cofinality (e.g., ). In particular, could be either or , where 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 . Some common examples are the following:Sets with greater cardinality
Sets with cardinality greater than include: *the set of all subsets of (i.e., power set ) *the set Power set#Representing subsets as functions, 2R of indicator functions defined on subsets of the reals (the set is isomorphic to – the indicator function chooses elements of each subset to include) *the set of all functions from to *the Lebesgue measure, Lebesgue σ-algebra of , i.e., the set of all Lebesgue measurable sets in . *the set of all Lebesgue integration, Lebesgue-integrable functions from to *the set of all Measurable function, Lebesgue-measurable functions from to *the Stone–Čech compactifications of , and *the set of all automorphisms of the (discrete) field of complex numbers. These all have cardinality (Beth number#Beth two, beth two).References
Bibliography
*Paul Halmos, ''Naive set theory''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition). *Thomas Jech, Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. . *Kenneth Kunen, Kunen, Kenneth, 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. . {{PlanetMath attribution, urlname=CardinalityOfTheContinuum, title=cardinality of the continuum Cardinal numbers Set theory Infinity