How did Georg Cantor show that some infinite sets are larger than others?
Georg Cantor, the founder of modern set theory, demonstrated that infinite sets can have different sizes by comparing their cardinalities through one-to-one correspondence.1
While some infinite sets, such as the rational numbers and integers, can be paired off and are considered "countably infinite," Cantor proved that other sets, such as the real numbers, are "uncountable" and strictly larger. He famously used a diagonalization argument to show that it is impossible to establish a one-to-one correspondence between these sets.12
Furthermore, Cantor’s theorem establishes that the power set of any set—the collection of all its subsets—always has a strictly greater cardinality than the original set. This discovery led to the creation of an infinite hierarchy of transfinite numbers, denoted by the Hebrew letter aleph (), representing sizes of infinity that grow progressively larger.12