Countable set Totally Explained

Everything about Countable Set totally explained

In mathematics, a countable set is a set with the same cardinality (for example, number of elements) as some subset of the set of natural numbers. The term was originated by Georg Cantor; it stems from the fact that the natural numbers are often called counting numbers. A set that isn't countable is called uncountable.
Some authors use countable set to mean a set with exactly as many elements as the set of natural numbers. The difference between the two definitions is that under the former, finite sets are also considered to be countable, while under the latter definition, they're not considered to be countable. To resolve this ambiguity, the term at most countable is sometimes used for the former notion, and countably infinite for the latter.

Definition

A set S is called countable if there exists an injective function ť

fcolon S 	o mathbb such that

c = 0 if a/b ≥ 0 and c = 1 otherwise.

0 maps to (0,1,0)

1 maps to (1,1,0)

−1 maps to (1,1,1)

1/2 maps to (1,2,0)

−1/2 maps to (1,2,1)

2 maps to (2,1,0)

−2 maps to (2,1,1)

1/3 maps to (1,3,0)

−1/3 maps to (1,3,1)

3 maps to (3,1,0)

−3 maps to (3,1,1)

1/4 maps to (1,4,0)

−1/4 maps to (1,4,1)

2/3 maps to (2,3,0)

−2/3 maps to (2,3,1)

3/2 maps to (3,2,0)

−3/2 maps to (3,2,1)

4 maps to (4,1,0)

−4 maps to (4,1,1)

...

By a similar development, the set of algebraic numbers is countable, and so is the set of definable numbers. THEOREM: (Assuming the axiom of countable choice) The union of countably many countable sets is countable.
For example, given countable sets a, b, c ...

Using a variant of the triangular enumeration we saw above:

a₀ maps to 0

a₁ maps to 1

b₀ maps to 2

a₂ maps to 3

b₁ maps to 4

c₀ maps to 5

a₃ maps to 6

b₂ maps to 7

c₁ maps to 8

d₀ maps to 9

a₄ maps to 10

...

Note that this only works if the sets a, b, c,... are disjoint. If not, then the union is even smaller and is therefore also countable by a previous theorem. THEOREM: The set of all finite-length sequences of natural numbers is countable.
This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which is a countable set (finite Cartesian product). So we're talking about a countable union of countable sets, which is countable by the previous theorem. THEOREM: The set of all finite subsets of the natural numbers is countable.
If you've a finite subset, you can order the elements into a finite sequence. There are only countably many finite sequences, so also there are only countably many finite subsets.

Further theorems about uncountable sets

The set of real numbers is uncountable (see Cantor's first uncountability proof), and so is the set of all sequences of natural numbers and the set of all subsets of N (see Cantor's diagonal argument).

The minimal model of set theory is countable

If there's a set which is a standard model (see inner model) of ZFC set theory, then there's a minimal standard model (see Constructible universe). The Löwenheim-Skolem theorem can be used to show that this minimal model is countable. The fact that the notion of "uncountability" makes sense even in this model, and in particular that this model M contains elements which are

subsets of M, hence countable,

but uncountable from the point of view of M, was seen as paradoxical in the early days of set theory, see Skolem's paradox. The minimal standard model includes all the algebraic numbers and all effectively computable transcendental numbers, as well as many other kinds of numbers.

Total orders

Countable sets can be totally ordered in various ways, for example:

well orders (see also ordinal number):

the usual order of natural numbers
the integers in the order 0, 1, 2, 3, .., -1, -2, -3, ..

other:

the usual order of integers
the usual order of rational numbers

Further Information

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