nLab
infinite set

Infinite sets

Idea

A set is infinite if it is not finite.

The existence of an infinite set is usually given by an axiom of infinity. The main example is the set of natural numbers.

Definitions

As you can see from finite set, there are at least five definitions of that term, which are all equivalent given the axiom of choice. The negation of any of these gives a definition of infinite set.

However, the definition usually used in practice in constructive mathematics is this:

Definition

A set SS is infinite if, given any natural number nn and a finite sequence (x 1,,x n)(x_1, \ldots, x_n) of elements of SS, there exists an element yy of SS such that y=x iy = x_i is always false.

In other words, given any function ff from a Kuratowski-finite set to SS, there exists an element of SS that is not in the image of ff. This is essentially a variation of Richard Dedekind's definition of a Dedekind-infinite set.

Note that you can make this definition work without previously assuming the existence of natural numbers, by using an infinity-free definition of Kuratowski-finite set.

Remarks

Probably a lot to say about the relation between the various definitions of infinite set (the one above, the negations of the definitions of finite set, and others that might be studied). In the meantime, try the English Wikipedia.

Revised on January 16, 2011 03:15:13 by Toby Bartels (75.117.106.4)