4.7 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Continuum hypothesis | 2/5 | https://en.wikipedia.org/wiki/Continuum_hypothesis | reference | science, encyclopedia | 2026-05-05T09:59:15.765779+00:00 | kb-cron |
Two sets are said to have the same cardinality or cardinal number if there exists a bijection (a one-to-one correspondence) between them. Intuitively, for two sets
S
{\displaystyle S}
and
T
{\displaystyle T}
to have the same cardinality means that it is possible to "pair off" elements of
S
{\displaystyle S}
with elements of
T
{\displaystyle T}
in such a fashion that every element of
S
{\displaystyle S}
is paired off with exactly one element of
T
{\displaystyle T}
and vice versa. Hence, the set
{
banana
,
apple
,
pear
}
{\displaystyle \{{\text{banana}},{\text{apple}},{\text{pear}}\}}
has the same cardinality as
{
yellow
,
red
,
green
}
{\displaystyle \{{\text{yellow}},{\text{red}},{\text{green}}\}}
despite the sets themselves containing different elements. With infinite sets such as the set of integers or rational numbers, the existence of a bijection between two sets becomes more difficult to demonstrate. The rational numbers
Q
{\displaystyle \mathbb {Q} }
seemingly form a counterexample to the continuum hypothesis: the integers form a proper subset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rational numbers than integers and more real numbers than rational numbers. However, this intuitive analysis is flawed since it does not take into account the fact that all three sets are infinite. Perhaps more importantly, it in fact conflates the concept of "size" of the set
Q
{\displaystyle \mathbb {Q} }
with the order or topological structure placed on it. In fact, it turns out the rational numbers can actually be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size (cardinality) as the set of integers: they are both countable sets. Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers (see Cantor's first uncountability proof and Cantor's diagonal argument). His proofs, however, give no indication of the extent to which the cardinality of the integers is less than that of the real numbers. Cantor proposed the continuum hypothesis as a possible solution to this question. In simple terms, the Continuum Hypothesis (CH) states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. That is, every infinite set
S
⊆
R
{\displaystyle S\subseteq \mathbb {R} }
of real numbers can either be mapped one-to-one into the integers or the real numbers can be mapped one-to-one into
S
{\displaystyle S}
. Since the real numbers are equinumerous with the powerset of the integers, i.e.
|
R
|
=
2
ℵ
0
{\displaystyle |\mathbb {R} |=2^{\aleph _{0}}}
, CH can be restated as follows:
Assuming the axiom of choice, there is a unique smallest cardinal number
ℵ
1
{\displaystyle \aleph _{1}}
greater than
ℵ
0
{\displaystyle \aleph _{0}}
, and the continuum hypothesis is in turn equivalent to the equality
2
ℵ
0
=
ℵ
1
{\displaystyle 2^{\aleph _{0}}=\aleph _{1}}
.