<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.

(01 Mar 1995)

1. Unlimited or boundless, in time or space; as, infinite duration or distance. "Whatever is finite, as finite, will admit of no comparative relation with infinity; for whatever is less than infinite is still infinitely distant from infinity; and lower than infinite distance the lowest or least can not sink." (H. Brooke)

2. Without limit in power, capacity, knowledge, or excellence; boundless; immeasurably or inconceivably great; perfect; as, the infinite wisdom and goodness of God; opposed to finite. "Great is our Lord, and of great power; his understanding is infinite." (Ps. Cxlvii. 5) "O God, how infinite thou art!" (I. Watts)

3. Indefinitely large or extensive; great; vast; immense; gigantic; prodigious. "Infinite riches in a little room." (Marlowe) "Which infinite calamity shall cause To human life." (Milton)

4. <mathematics> Greater than any assignable quantity of the same kind; said of certain quantities.

5. Capable of endless repetition; said of certain forms of the canon, called also perpetual fugues, so constructed that their ends lead to their beginnings, and the performance may be incessantly repeated.

Synonyms: Boundless, immeasurable, illimitable, interminable, limitless, unlimited, endless, eternal.

Origin: L. Infinitus: cf. F. Infini. See In- not, and Finite.

1. That which is infinite; boundless space or duration; infinity; boundlessness. "Not till the weight is heaved from off the air, and the thunders roll down the horizon, will the serene light of God flow upon us, and the blue infinite embrace us again." (J. Martineau)

2. <mathematics> An infinite quantity or magnitude.

3. An infinity; an incalculable or very great number. "Glittering chains, embroidered richly o'er With infinite of pearls and finest gold." (Fanshawe)

4. The Infinite Being; God; the Almighty.

(01 Mar 1998)