How can a set be infinite even though it can be paired with a smaller-looking set?

In set theory, an infinite set can be paired with a proper subset of itself because, unlike finite sets, infinite sets do not follow the intuitive rule that a whole must be larger than its parts.12

This phenomenon occurs because the size, or cardinality, of an infinite set is defined by the ability to establish a one-to-one correspondence between its elements and those of another set. For example, the set of natural numbers ($\mathbb{N}$) can be paired with the set of its squares ($\{n^2[3][2]|n\in \mathbb{N}\}$) by matching each number $n$ with $n^2$. Even though the squares are a proper subset of the natural numbers, the two sets are equivalent because every element in one can be uniquely paired with an element in the other with no leftovers.23

This property was famously noted by Galileo Galilei and later used by Richard Dedekind to define an infinite set as one that can be put into a one-to-one relationship with a proper subset of itself.12