In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers
Contents
- Uncountability
- Cardinal equalities
- Alternative explanation for c 2 0 displaystyle mathfrak c2aleph 0
- Beth numbers
- The continuum hypothesis
- Sets with cardinality of the continuum
- Sets with greater cardinality
- References
The real numbers
This was proven by Georg Cantor in his 1874 uncountability proof, part of his groundbreaking study of different infinities, and later more simply in his diagonal argument. 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 with
The smallest infinite cardinal number is
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.
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.
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.
- Define a map
f : R → ℘ ( Q ) from the reals to the power set of the rationals,Q by sending each real numberx to the set{ q ∈ Q : q ≤ x } of all rationals less than or equal tox (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 thatc ≤ 2 ℵ 0 - Let
{ 0 , 2 } N { 0 , 2 } . This set has cardinality2 ℵ 0 ℘ ( N ) is given by the indicator function). Now associate to each such sequence( a i ) i ∈ N [ 0 , 1 ] with the ternary-expansion given by the digitsa 1 , a 2 , … , i.e.,∑ i = 1 ∞ a i 3 − i i -th digit after the fractional point isa i 3 . 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 that2 ℵ 0 ≤ c .
By the Cantor–Bernstein–Schroeder theorem we conclude that
The cardinal equality
By using the rules of cardinal arithmetic one can also show that
where n is any finite cardinal ≥ 2, and
where
Alternative explanation for c = 2 ℵ 0 {displaystyle {mathfrak {c}}=2^{aleph _{0}}}
Every real number has at least one infinite decimal expansion. For example,
1/2 = 0.50000...1/3 = 0.33333...(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
Since each real number can be broken into an integer part and a decimal fraction, we get
since
On the other hand, if we map
and thus
Beth numbers
The sequence of beth numbers is defined by setting
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):
The continuum hypothesis
The famous continuum hypothesis asserts that
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
Sets with cardinality of the continuum
A great many sets studied in mathematics have cardinality equal to
Sets with greater cardinality
Sets with cardinality greater than
These all have cardinality