How did Richard Dedekind define an infinite set using one-to-one correspondence?

In his 1888 work Was sind und was sollen die Zahlen? ("What Are and Should Be the Numbers?"), the German mathematician Richard Dedekind provided a precise definition of an infinite set based on the concept of one-to-one correspondence.1

Dedekind defined a set as infinite if and only if the entire set can be put into a one-to-one correspondence with a proper part (a subset) of itself. This characterization distinguishes infinite sets from finite sets, as finite sets cannot be mapped onto a proper subset in this manner. By utilizing this method—which had been previously explored by Bernhard Bolzano—Dedekind contributed significantly to the modern mathematical understanding of the infinite.12