Countable set

A countable set is an infinite set whose elements can be numbered by natural numbers . More formal: many $X$ ${\ displaystyle X}$ $X$ is countable if there is a bijection with many natural numbers: $X\leftrightarrow \mathbb {N}$ ${\ displaystyle X \ leftrightarrow \ mathbb {N}}$ ${\ displaystyle X \ leftrightarrow \ mathbb {N}}$ in other words, a countable set is a set equipotent to the set of natural numbers. In the hierarchy of Alephs, the power of a countable set is denoted by $\aleph _{0}$ ${\ displaystyle \ aleph _ {0}}$ $\ aleph _ {0}$ ("Aleph-null").

A countable set is a “simplest” infinite set in the following sense: in any infinite set there is a countable subset; every subset of a countable set is finite or countable; if we attach a finite or countable to an infinite set, then we get a set equipotent with the original ^[1] .

Combinations of a finite or countable number of countable sets, as well as direct products of a finite number of countable sets, are countable ^[2] ^[1] . The set of all finite subsets of a countable set is countable; however, the set of all subsets of a countable set is continuous , and is not countable.

Countable are sets of natural numbers $\mathbb {N}$ ${\ displaystyle \ mathbb {N}}$ $\ mathbb {N}$ , integers $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ $\ mathbb {Z}$ rational numbers $\mathbb {Q}$ ${\ displaystyle \ mathbb {Q}}$ $\ mathbb {Q}$ algebraic numbers $\mathbb {A}$ ${\ displaystyle \ mathbb {A}}$ ${\ displaystyle \ mathbb {A}}$ . Countable are objects obtained as a result of recursive procedures , in particular, these are computable numbers , arithmetic numbers (as a result, the ring of periods is countable, since each period is computable ). A countable set of all finite words over a countable alphabet and a set of all words over a finite alphabet. Any objects that can be defined with a one-to-one comparison with a countable set are countable, for example: any infinite family of disjoint open intervals on the real axis; the set of all lines on the plane , each of which contains at least two points with rational coordinates ; any infinite number of points on the plane, all pairwise distances between the elements of which are rational.

An uncountable set is such an infinite set that is not countable, such, in particular, are the sets of real numbers $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ $\ mathbb {R}$ complex numbers $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ ${\ displaystyle \ mathbb {C}}$ Cayley numbers $\mathbb {O}$ ${\ displaystyle \ mathbb {O}}$ $\ mathbb O$ . Thus, any set is either finite, countable, or uncountable.

Notes

↑ ¹ ² Brudno, 1971 , p. fourteen.
↑ V. A. Ilyin , V. A. Sadovnichy , Bl. H. Sendov . Chapter 2. Real numbers // Mathematical analysis / Ed. A.N. Tikhonova . - 3rd ed., Revised. and add. - M .: Prospect, 2006. - T. 1. - S. 62-63. - 672 p. - ISBN 5-482-00445-7 .

Literature

Brudno A. L. The theory of functions of a real variable. - M .: Nauka, 1971. - 119 p.

[_46c663f65fc6a43f-1] ¹ ² Brudno, 1971 , p. fourteen.

[ilyin-2] V. A. Ilyin , V. A. Sadovnichy , Bl. H. Sendov . Chapter 2. Real numbers // Mathematical analysis / Ed. A.N. Tikhonova . - 3rd ed., Revised. and add. - M .: Prospect, 2006. - T. 1. - S. 62-63. - 672 p. - ISBN 5-482-00445-7 .

Countable set

Notes

Literature

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