Changes

A [[set]] X is '''countable''' if and only if there is a [[bijection]] from X to the set of [[Natural Numbers|natural numbers]].  Countable sets include the set of [[integer]]s and the set of [[rational number]]s.

A [[set]] X is '''countable''' if and only if there is a [[bijection]] from X to the set of [[Natural Numbers|natural numbers]].  Countable sets include the set of [[integer]]s and the set of [[rational number]]s. Countable sets are said to have [[cardinality]] <math>\aleph</math>.

 

There is no smallest countable set. Indeed, the set of natural numbers is in bijection with the natural numbers without 0: subtracting 1 from every number gives a bijection from the first set to the second. Repeating this process shows that for any initial segment of the the natural numbers (such as {1, 2, ..., n}), we have a bijection between the set of natural numbers and the set of natural numbers without this segment. The bijection is simply subraction by n.

[[Category:Set theory]]

[[Category:Set theory]]