# infinite

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Related to infinitely: Infinitely small

## infinite

*Maths*

**a.**having an unlimited number of digits, factors, terms, members, etc.

**b.**(of a set) able to be put in a one-to-one correspondence with part of itself

**c.**(of an integral) having infinity as one or both limits of integration

## infinite

[′in·fə·nət] (mathematics)

Larger than any fixed number.

## infinite

(mathematics)1. Bigger than any natural number. There are
various formal set definitions in set theory: a set X is
infinite if

(i) There is a bijection between X and a proper subset of X.

(ii) There is an injection from the set N of natural numbers to X.

(iii) There is an injection from each natural number n to X.

These definitions are not necessarily equivalent unless we accept the Axiom of Choice.

2. The length of a line extended indefinitely.

See also infinite loop, infinite set.

(i) There is a bijection between X and a proper subset of X.

(ii) There is an injection from the set N of natural numbers to X.

(iii) There is an injection from each natural number n to X.

These definitions are not necessarily equivalent unless we accept the Axiom of Choice.

2. The length of a line extended indefinitely.

See also infinite loop, infinite set.

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