infinite set

Infinite sets

Infinite sets


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.


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:


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. (Although because only the image matters, the definition would be equivalent if we required SS to be Bishop-finite, that is finite in the strictest sense.) 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.


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.

Last revised on August 19, 2019 at 00:32:09. See the history of this page for a list of all contributions to it.