Aleph Null

Aleph Null

The symbol א‎₀ (aleph with a subscript of 0). Refers to the cardinality of a countably infinite set.

Note: Aleph (א) is the first letter of the Hebrew alphabet.

Key Formula

|\mathbb{N}| = \aleph_0

Where:

$\mathbb{N}$ = The set of natural numbers {1, 2, 3, …}
$|\mathbb{N}|$ = The cardinality (size) of the set of natural numbers
$\aleph_0$ = Aleph null, the cardinality of any countably infinite set

Example

Problem: Show that the set of even positive integers {2, 4, 6, 8, …} has the same cardinality as the set of all natural numbers {1, 2, 3, 4, …}, and therefore also has cardinality ℵ₀.

Step 1: Define a function that pairs every natural number with exactly one even number. Let f(n) = 2n.

f(n) = 2n

Step 2: Check the pairing: 1 maps to 2, 2 maps to 4, 3 maps to 6, and so on. Every natural number is paired with a unique even number, and every even number is reached.

1 \to 2, \quad 2 \to 4, \quad 3 \to 6, \quad 4 \to 8, \quad \ldots

Step 3: Because this function is a one-to-one correspondence (a bijection), the two sets have the same cardinality.

|\{2, 4, 6, 8, \ldots\}| = |\mathbb{N}| = \aleph_0

Answer: The set of even positive integers is countably infinite, so its cardinality is ℵ₀ — the same as the natural numbers, even though the evens are a proper subset of the naturals.

Another Example

Problem: Show that the set of integers {…, −2, −1, 0, 1, 2, …} also has cardinality ℵ₀.

Step 1: List the integers in a sequence that covers every integer exactly once: 0, 1, −1, 2, −2, 3, −3, …

0, 1, -1, 2, -2, 3, -3, \ldots

Step 2: This listing creates a bijection between the natural numbers and the integers: pair the 1st natural number with 0, the 2nd with 1, the 3rd with −1, and so on.

1 \to 0, \quad 2 \to 1, \quad 3 \to -1, \quad 4 \to 2, \quad 5 \to -2, \quad \ldots

Step 3: Since every integer appears exactly once in this list, a one-to-one correspondence exists.

|\mathbb{Z}| = \aleph_0

Answer: The set of all integers has cardinality ℵ₀, the same as the natural numbers.

Frequently Asked Questions

Is aleph null the biggest infinity?

No — aleph null is the smallest infinity. Georg Cantor proved that the set of real numbers is strictly larger than any countably infinite set. Its cardinality is a larger infinite cardinal number. In fact, there is an entire hierarchy of larger infinities:

\aleph_1, \aleph_2, \ldots

Is aleph null a number you can use in arithmetic?

Aleph null is a cardinal number, but it does not follow the usual rules of arithmetic. For instance,

\aleph_0 + 1 = \aleph_0

and

\aleph_0 + \aleph_0 = \aleph_0

. Adding or multiplying finite amounts to it does not change it. This is part of what makes infinite cardinal arithmetic different from ordinary arithmetic.

Aleph Null (ℵ₀) vs. Infinity (∞)

Aleph null is a precise cardinal number that measures the size of countably infinite sets. The infinity symbol ∞ is used more loosely — for example, in limits (

\lim_{x \to \infty}

) — and does not by itself specify which 'size' of infinity is meant. You can think of

\aleph_0

as a specific, well-defined level of infinity, while ∞ is a general concept meaning 'without bound.'

Why It Matters

Aleph null is the foundation of Cantor's theory of infinite sets, which reshaped modern mathematics. It gives a precise way to say that some infinite collections (like the rationals) are the same size as the natural numbers, while others (like the real numbers) are strictly larger. Understanding

\aleph_0

is essential in set theory, logic, and any field that deals rigorously with infinite structures.

Common Mistakes

Mistake: Thinking that a proper subset of the natural numbers must be 'smaller' than the naturals.

Correction: Infinite sets can be put into one-to-one correspondence with proper subsets of themselves. The even numbers, for example, form a proper subset of the naturals yet have the same cardinality, ℵ₀.

Mistake: Assuming all infinite sets have cardinality ℵ₀.

Correction: Only countably infinite sets have cardinality ℵ₀. The set of real numbers, for instance, is uncountably infinite and has a strictly larger cardinality.

Related Terms

Cardinality — The concept that ℵ₀ measures for infinite sets
Countably Infinite — Any countably infinite set has cardinality ℵ₀
Set — The collection whose size ℵ₀ describes
Infinite — ℵ₀ is the smallest infinite cardinal
Infinity — General concept; ℵ₀ is a specific level
Finite — Contrasts with the infinite size ℵ₀
Cardinal Numbers — ℵ₀ is the first infinite cardinal number