Why do the natural numbers and integers have the same cardinality?

In mathematics, the natural numbers and integers have the same cardinality because they can be placed into a one-to-one correspondence, meaning their elements can be paired perfectly without omission or repetition.123

Both sets are considered countably infinite, possessing a cardinality known as aleph-null (${\aleph }_0$). While it may seem counterintuitive that the set of integers (which includes negative numbers and zero) is not "larger" than the set of natural numbers, this is a fundamental property of infinite sets.143

To demonstrate this correspondence, one can list the integers in a sequence that pairs them with the natural numbers:
* 0 corresponds to 1
* 1 corresponds to 2
* -1 corresponds to 3
* 2 corresponds to 4
* -2 corresponds to 5
* ...and so on.

Because every integer can eventually be assigned a unique position in this never-ending list, the two sets are equivalent in size. This concept, pioneered by Georg Cantor, illustrates that infinite sets can have the same cardinality even when one appears to be a subset of the other or contains additional elements.123