Why are transcendental numbers more numerous than integers even though both collections are infinite?

To understand why transcendental numbers are more numerous than integers, it is necessary to clarify that both collections are infinite, but they possess different "sizes" or cardinalities.12

The set of integers is countably infinite, meaning its elements can be put into a one-to-one correspondence with the natural numbers. This size is denoted as aleph-null (${\aleph }_0$).13

In contrast, transcendental numbers are a subset of the real numbers. Georg Cantor proved that the set of all real numbers is uncountable, meaning it is strictly larger than the set of integers. Because the set of algebraic numbers (those that are roots of polynomial equations with rational coefficients) is also countable, the vast majority of real numbers must be transcendental.14567

Essentially, while integers can be listed and paired with natural numbers, the real numbers—and by extension, the transcendental numbers—cannot be paired in this way, representing a higher order of infinity known as the continuum (c).135