Why aleph‑null + aleph‑null = aleph‑null (The Math of Infinity)

Aleph-Null: The Smallest Infinity

Aleph-null (ℵ₀) is a cardinal number in set theory that represents the cardinality, or size, of the set of natural numbers {1, 2, 3, …}. It is the first transfinite cardinal number and is used to describe the size of infinite sets.

Arithmetic with Aleph-Null

Aleph-null is closed under addition, multiplication, and exponentiation. For example:

ℵ₀ + ℵ₀ = ℵ₀

ℵ₀ × ℵ₀ = ℵ₀

ℵ₀^ℵ₀ = ℵ₀

Comparing Infinite Sets

Aleph-null can also be used to compare the sizes of infinite sets. For example, the set of rational numbers has the same cardinality as the set of natural numbers (ℵ₀), while the set of real numbers has a strictly larger cardinality of 2^ℵ₀.

Cantor’s Diagonal Argument

Aleph-null plays a key role in Cantor’s diagonal argument, which proves that the set of real numbers has a larger cardinality than the set of natural numbers — meaning not all infinities are equal.