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Aleph-null aleph-null, often written as $\aleph_0 \aleph_0$, represents the cardinality of the Cartesian product of the set of natural numbers with itself. The set of natural numbers, denoted by $\mathbb{N}$, is the set $\{0, 1, 2, 3, \dots\}$. Its cardinality, the "number of elements" in the set, is aleph-null ($\aleph_0$).

The Cartesian product $\mathbb{N} \times \mathbb{N}$ is the set of all ordered pairs $(a, b)$ where $a$ and $b$ are natural numbers. Examples of elements in this set include $(0, 0)$, $(1, 5)$, and $(100, 23)$.

A fundamental result in set theory is that the cardinality of $\mathbb{N} \times \mathbb{N}$ is equal to the cardinality of $\mathbb{N}$ itself. This means that $\aleph_0 \aleph_0 = \aleph_0$. This result might seem counterintuitive, as one might expect that doubling the "size" of a set would result in a larger cardinality.

This equality demonstrates that aleph-null is a countably infinite cardinality. The proof of this involves establishing a bijection (a one-to-one correspondence) between the elements of $\mathbb{N} \times \mathbb{N}$ and the elements of $\mathbb{N}$. One common method to construct such a bijection is by using a diagonalization argument, similar to Cantor's pairing function, which systematically enumerates all pairs of natural numbers. This property, that $\aleph_0 \times \aleph_0 = \aleph_0$, is a key characteristic of countably infinite sets.