Infinite set

An infinite set is a set that is not finite: its elements cannot be put in one-to-one correspondence with any finite set of natural numbers.

Set: A collection of distinct objects treated as a single mathematical object. An infinite set is a set whose elements cannot be counted to a finite total.
Cantor's diagonal argument: A proof method that constructs an object differing from every item in a proposed enumeration. It proves that the real numbers cannot be listed, establishing an uncountable infinite set.
Finite set: A set whose elements can be placed in one-to-one correspondence with a natural number’s finite initial segment. Finite sets lack the self-matching behavior characteristic of infinite sets.
Infinite cardinal arithmetic: Arithmetic operations on cardinal numbers that describe the sizes of infinite sets. It explains results such as an infinite set having the same size as two copies of itself.
Cardinality: A measure of the number of elements in a set, including infinite sets. Cardinality distinguishes sizes of infinite sets that cannot be counted finitely.
Cantor's theorem: The theorem that every set has strictly smaller cardinality than its power set. It shows that no infinite set is as large as its collection of subsets.
Natural numbers: The positive whole numbers, often including zero by convention, used for counting and ordering. They form the standard example of a countably infinite set.
Axiom of choice: The set-theoretic principle that a choice function exists for every family of nonempty sets. It affects whether certain infinite sets and their subsets can be constructed or well-ordered.
Bijection: A function that is both one-to-one and onto, pairing every element of its domain with exactly one element of its codomain. Bijections define when two infinite sets have the same size.
Hilbert's paradox of the Grand Hotel: A thought experiment in which a fully occupied countably infinite hotel can still accommodate more guests. Its room-shifting arguments illustrate how an infinite set can match a proper subset.

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