Cardinals · Definition

Infinite set

set theorycardinals

Before you start you need to know

Infinite sets are those that never run out when counted. Studying them reveals that there are different sizes of infinity.

Definition

A set \(A\) is infinite if it is not finite, that is, if there is no bijection between \(A\) and \(\{1,\dots,n\}\) for any \(n\in\mathbb{N}\).

\[A\not\sim\{1,\dots,n\}\ \ \forall n\in\mathbb{N}\]

Examples: \(\mathbb{N}\), \(\mathbb{Z}\) and \(\mathbb{R}\) are infinite. For \(\mathbb{N}\), take \(f(n)=n+1\): it is injective but not surjective, because it never reaches \(0\). A finite set admits no injection into a proper subset of itself, so \(\mathbb{N}\) is not finite.

Properties: every subset of a finite set is finite. Equivalently, if \(A\subseteq B\) and \(A\) is infinite, then \(B\) is infinite. Moreover, the union of an infinite set with a finite one is still infinite.

Typical mistakes: thinking that infinite means «very large». The set \(\{x\in\mathbb{N}: x\le 10^{100}\}\) is finite, however huge. Not all infinities are equal either: \(\mathbb{N}\) and \(\mathbb{R}\) cannot be put in bijection, as Cantor's diagonal argument shows.

Think about it

Why is being large not enough for a set to be infinite?

Because infinite means not finite: there is no bijection with any \(\{1,\dots,n\}\), whatever its size.

Exercises 10

  1. How is an infinite set defined?
  2. Which of these sets is infinite?
  3. If \(A\subseteq B\) and \(A\) is infinite, what follows about \(B\)?
  4. Can an infinite set be a subset of a finite set?
  5. Is \([0,1]\) a finite set?
  6. If \(A\) is infinite, is there a bijection between \(A\) and \(\{1,\dots,n\}\) for some \(n\in\mathbb{N}\)?
  7. Let \(A\) be finite and \(B\) infinite. What can be said about \(A\cup B\)?
  8. Is the set of even natural numbers infinite?
  9. Let \(f(n)=n+1\) from \(\mathbb{N}\) to \(\mathbb{N}\). Why does it prove that \(\mathbb{N}\) is infinite?
  10. Let \(A\) be infinite and \(a\in A\). Is \(A\setminus\{a\}\) infinite?

What comes next

References

  • Herbert B. Enderton, Elements of Set Theory, 1st edition, Academic Press, 1977.
  • Thomas Jech, Set Theory (The Third Millennium Edition), 3rd millennium edition, Springer, 2003.