Cardinality of the continuum
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, | A | < 2 | A | {\displaystyle |A|<2^{|A|}} (and so that the power set ℘ (N) {\displaystyle \wp (\mathbb {N})} of the natural numbers N {\displaystyle \mathbb {N} } is uncountable). In fact, the cardinality of ℘ (N) {\displaystyle \wp (\mathbb {N})}, by definition 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}}, is equal to c {\displaystyle {\mathfrak {c}}}. 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. In the other direction, the binary expansions of numbers in the half-open interval [ 0, 1) {\displaystyle [0,1)}, 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. This can be made into a one-to-one mapping by that adds one to the non-terminating repeating-1 expansions, mapping them into [ 1, 2) {\displaystyle [1,2)}. Thus, we conclude that

The cardinal equality c 2 = c {\displaystyle {\mathfrak {c}}^{2}={\mathfrak {c}}} 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 2 c {\displaystyle 2^{\mathfrak {c}}} is the cardinality of the power set of R, and 2 c > c {\displaystyle 2^{\mathfrak {c}}>{\mathfrak {c}}}.
detectable_content:postscript