Countably infinite

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A set is countably infinite if there exists a one to one mapping between it and the natural numbers.

For example, the set containing all even numbers is countably infinite as displayed by the mapping:

  • 0 maps to 0
  • 2 maps to 1
  • 4 maps to 2
  • ...
  • N maps to (N/2) where N is an even number

Likewise, pairs of natural numbers are countably infinite through the following mapping:

  • (0,0) maps to 0
  • (0,1) maps to 1
  • (1,0) maps to 2
  • (0,2) maps to 3
  • (1,2) maps to 4
  • (2,0) maps to 5
  • ...
  • (m,n) maps to 0.5*k*(k+1)+m, where k=m+n

Sometimes it is necessary to use more than one mapping. This is where you map the set you want to show as countably infinite to another set. You then map this other set to the natural numbers. For example, the Rational numbers can easily be mapped to (a subset of) the pairs of integers through p/q maps to (p,q).


There are other, uncountable infinites. It is easy to show, for instance, that the set of real numbers is not countably infinite, using Cantor's diagonal argument. The size of the countable infinite sets is the smallest infinity.