Uncountable set

An uncountable set is an infinite set that is not countable .

Some equivalent definitions of uncountability for a set $X$ ${\ displaystyle X}$ $X$ :

there is no injective mapping $X$ ${\ displaystyle X}$ $X$ into many natural numbers $\mathbb {N}$ ${\ displaystyle \ mathbb {N}}$ $\ mathbb {N}$ ;
$X$ {\ displaystyle X} $X$ not empty , and for each numbered sequence of elements $X$ {\ displaystyle X} $X$ there is at least one element $X$ {\ displaystyle X} $X$ not entering into it;
- in other words: $X$ ${\ displaystyle X}$ $X$ is nonempty and there is no surjective mapping of the set of natural numbers $\mathbb {N}$ ${\ displaystyle \ mathbb {N}}$ $\ mathbb {N}$ on $X$ ${\ displaystyle X}$ $X$ ;
power $X$ ${\ displaystyle X}$ $X$ is neither finite nor equal $\aleph _{0}$ ${\ displaystyle \ aleph _ {0}}$ $\ aleph_0$ .

These definitions are equivalent in the Zermelo - Frenkel system without using the axiom of choice . The proof of the equivalence of these definitions with the following:

power $X$ ${\ displaystyle X}$ $X$ strictly exceeds $\aleph _{0}$ ${\ displaystyle \ aleph _ {0}}$ $\ aleph_0$

- requires the attraction of the axiom of choice.

A subset of an uncountable set is uncountable. The simplest example of an uncountable set is the continuum ; the question of the existence of uncountable sets with cardinality less than the cardinality of the continuum is the content of the continuum hypothesis .

Literature

M.I. Wojciechowski. Countless // Mathematical Encyclopedia / I. M. Vinogradov (Chap. Ed.). - M .: Soviet Encyclopedia, 1982. - T. 3. - 592 p. - 150,000 copies.

Uncountable set

Literature

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