Dedekind Infinite

A set is called Dedekind infinite if it can be put into a one-to-one correspondence with a proper subset of itself—that is, you can match each element of the set with a unique element of a smaller part of itself, leaving some elements out, yet still pairing everything up without leftovers. This property shows the set is "infinite" in a strong sense because it can be "shrunken" and still resemble itself perfectly. For example, the set of natural numbers (1, 2, 3, ...) is Dedekind infinite because you can pair each number with the next (1 with 2, 2 with 3, etc.), skipping some elements still leaves a similar structure.