Cardinal Numbers

Cardinal Numbers

The numbers 1, 2, 3, . . . as well as some types of infinity. Cardinal numbers are used to describe the number of elements in either finite or infinite sets.

Key Formula

|A| = n

Where:

$|A|$ = The cardinality (cardinal number) of set A
$n$ = The number of elements in set A

Worked Example

Problem: Find the cardinal number of the set A = {2, 5, 8, 11, 14}.

Step 1: List every element of the set: 2, 5, 8, 11, 14.

A = \{2,\; 5,\; 8,\; 11,\; 14\}

Step 2: Count the elements. There are 5 distinct elements in the set.

|A| = 5

Step 3: The cardinal number of A is 5. This means the set has a size, or cardinality, of 5.

Answer: The cardinal number of set A is 5.

Another Example

Problem: Compare the cardinal numbers of set B = {a, b, c} and set C = {10, 20, 30}.

Step 1: Count the elements of B: a, b, c — there are 3.

|B| = 3

Step 2: Count the elements of C: 10, 20, 30 — there are 3.

|C| = 3

Step 3: Both sets have the same cardinal number, even though their elements are completely different. Two sets with the same cardinality are said to have a one-to-one correspondence between their elements.

|B| = |C| = 3

Answer: Both sets have a cardinal number of 3. They are the same size despite containing different elements.

Frequently Asked Questions

What is the difference between cardinal numbers and natural numbers?

The natural numbers (1, 2, 3, …) are all cardinal numbers, and they measure the size of finite sets. However, cardinal numbers also extend beyond the natural numbers to describe the sizes of infinite sets. For example, ℵ₀ (aleph-null) is the cardinal number of the set of all natural numbers, and it is not itself a natural number.

Is zero a cardinal number?

Yes. Zero is the cardinal number of the empty set, since the empty set contains no elements. If A = {}, then |A| = 0.

Cardinal Numbers vs. Ordinal Numbers

Cardinal numbers answer 'how many?' — they measure the size of a set. Ordinal numbers answer 'in what position?' — they describe order or ranking (first, second, third, …). For example, if 5 runners finish a race, the cardinal number of runners is 5, while the runner who finishes in position 3 is described by the ordinal number 'third.' For finite sets, cardinals and ordinals behave similarly, but for infinite sets, they diverge in important ways.

Why It Matters

Cardinal numbers are foundational to set theory, which underpins nearly all of modern mathematics. They give us a precise way to compare the sizes of collections, even infinite ones. Georg Cantor's discovery that there are different sizes of infinity — that the set of real numbers has a strictly larger cardinal number than the set of natural numbers — was one of the most surprising results in the history of mathematics.

Common Mistakes

Mistake: Confusing cardinal numbers with ordinal numbers.

Correction: Cardinal numbers measure how many elements a set has (e.g., 'there are 5 students'). Ordinal numbers indicate position or rank (e.g., 'the 5th student'). They answer different questions.

Mistake: Assuming all infinite sets have the same cardinal number.

Correction: Not all infinities are equal. The set of natural numbers has cardinality ℵ₀, but the set of real numbers has a strictly larger cardinality. Cantor proved this with his famous diagonal argument.

Related Terms

Cardinality — The cardinality of a set is its cardinal number
Ordinal Numbers — Describe position rather than quantity
Set — Cardinal numbers measure the size of sets
Aleph Null — The smallest infinite cardinal number
Countably Infinite — Sets whose cardinality equals aleph-null
Uncountable — Sets with cardinality larger than aleph-null
Element of a Set — Cardinal numbers count elements in a set
Infinity — Infinite cardinals extend beyond finite counting