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What is the Cardinality of a Countably Infinite Set?

Published in Set Theory 2 mins read

The cardinality of a countably infinite set is ℵ₀ (aleph-null).

Understanding Cardinality and Countable Infinity

Cardinality refers to the "size" of a set. For finite sets, it's simply the number of elements. For infinite sets, it becomes more nuanced. A set is considered countably infinite if its elements can be put into a one-to-one correspondence with the set of natural numbers (1, 2, 3, ...). This means you can "count" the elements of the set, even though the counting will never end.

Aleph-Null (ℵ₀)

The symbol ℵ₀ (aleph-null) is used to denote the cardinality of countably infinite sets. It represents the "smallest" type of infinity. Examples of sets with cardinality ℵ₀ include:

  • The set of natural numbers (N)
  • The set of integers (Z)
  • The set of rational numbers (Q)

While these sets seem different, it's possible to create a one-to-one correspondence between each of them and the natural numbers, proving they have the same cardinality.

Examples of Countably Infinite Sets

Let's look at some examples to clarify:

  • Natural Numbers (N): This is the defining example. By definition, a set is countably infinite if it can be mapped to N.

  • Integers (Z): You might think there are "more" integers than natural numbers (because of the negative numbers), but you can create a mapping like this:
    1 <-> 0
    2 <-> 1
    3 <-> -1
    4 <-> 2
    5 <-> -2
    ...and so on. This shows that integers also have cardinality ℵ₀.

  • Rational Numbers (Q): Rational numbers are fractions, and it might seem impossible to list them in a way that corresponds to the natural numbers. However, a clever diagonalization argument proves that the rational numbers are also countably infinite.

Uncountable Sets

It's important to distinguish countably infinite sets from uncountable sets. An uncountable set is an infinite set that cannot be put into a one-to-one correspondence with the natural numbers. A famous example is the set of real numbers (R). The cardinality of the real numbers is denoted by c (for continuum) and is strictly greater than ℵ₀. Therefore, |R| > |N|, or c > ℵ₀.

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