How does a one-to-one correspondence compare the sizes of infinite sets?

In set theory, a one-to-one correspondence is the fundamental method for comparing the sizes of infinite sets. By attempting to pair every element of one set with exactly one element of another, mathematicians can determine if the sets have the same cardinality, or numerical size.12

If two infinite sets can be paired off completely with no elements left over from either, they are considered to have the same size. For example, the rational numbers can be placed in a one-to-one correspondence with the integers, meaning they share the same cardinality, known as aleph-null.12

Conversely, if it is mathematically proven that no such pairing can exist—as is the case with the real numbers compared to the integers—the sets are not of the same size. In such instances, the set that cannot be paired is considered larger. This approach, pioneered by Georg Cantor, revealed that there is not just one infinity, but an infinite hierarchy of transfinite numbers of increasing size.12