Ordinals and axioms · Definition

Construction of the naturals

set theoryordinals and axioms

Before you start you need to know

Before talking about sums or products, it helps to know where numbers come from. Set theory lets us define them using only membership.

Definition

The natural numbers are a set \(\mathbb{N}\) built inside set theory as finite ordinals, where each natural is the set of all naturals smaller than it.

\[n^+ = n \cup \{n\}\]

Define \(0 = \varnothing\) and, for each set \(n\), its successor \(n^+ = n \cup \{n\}\). Thus \(1 = \{0\}\), \(2 = \{0,1\}\), and in general \(n\) has exactly \(n\) elements. By the axiom of infinity there is an inductive set containing \(0\) closed under successor, and \(\mathbb{N}\) is the intersection of all such sets. This yields induction: every subset of \(\mathbb{N}\) containing \(0\) and closed under successor equals \(\mathbb{N}\).

Take \(3 = \{0,1,2\}\). Its successor is \(3^+ = \{0,1,2,3\}\), which has four elements. Each step adds exactly one new element, so counting is encoded in membership itself.

A common mistake is to treat \(\mathbb{N}\) as given without construction, or to confuse \(n\) with \(\{n\}\). Remember that \(1 = \{0\}\), not \(\{1\}\), and that \(n\) contains all naturals smaller than it.

Think about it

What are the elements of \(3\), and why is \(3\notin 3\)?

\(3=\{0,1,2\}\) has three elements, all less than 3; it does not contain 3 because its elements are strictly smaller.

Exercises 10

  1. What is the set \(1\) in the von Neumann construction?
  2. What set is \(0\) in this construction?
  3. How many elements does the natural \(4\) have?
  4. What is the successor of \(2\)?
  5. Compute \(2^+\) using the definition of successor.
  6. How many elements does \(3^+\) have?
  7. If \(n=\{0,1\}\), what is \(n^+\)?
  8. What is the set \(\mathbb{N}\cap\{0,1,2\}\) of naturals less than \(3\)?
  9. Let \(A=\{x\in\mathbb{N}: x\ne 0\}\). Which statement correctly shows that \(A\) is not inductive?
  10. Which statement gives a counterexample to the claim that \(n^+=n\) for every natural \(n\)?

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.