Ordinals and axioms · Definition

Construction of the reals

set theoryordinals and axioms

Before you start you need to know

After building \( \mathbb{Z} \) and \( \mathbb{Q} \) from \( \mathbb{N} \), we still need a field where every Cauchy sequence has a limit. Dedekind cuts fill the gaps left in \( \mathbb{Q} \).

Definition

The real numbers \( \mathbb{R} \) form a complete ordered field extending \( \mathbb{Q} \), in which every bounded Cauchy sequence converges. They are built from \( \mathbb{Q} \) using Dedekind cuts or Cauchy sequences.

\[\mathbb{R}=\{A\subseteq\mathbb{Q}: A \text{ es cortadura}\}\]

A Dedekind cut is a nonempty subset \( A\subseteq\mathbb{Q} \), different from \( \mathbb{Q} \), closed downward and without a maximum. Addition, multiplication and the order by inclusion are defined on these cuts, making \( \mathbb{R} \) a complete ordered field.

Take \( A=\{q\in\mathbb{Q}: q<0 \text{ or } q^2<2\} \). This cut represents \( \sqrt{2} \), which is not rational, so \( \mathbb{R} \) fills the gap that \( \mathbb{Q} \) left.

A typical mistake is confusing the cut with its endpoint: the cut \( A \) does not contain \( \sqrt{2} \) as an element, since \( \sqrt{2}\notin\mathbb{Q} \). The cut is the set of rationals lying below the number.

Think about it

Think about it: why does \( \sqrt{2} \) have no rational representative?

Because no rational has square equal to \( 2 \), so \( \mathbb{Q} \) has a gap there.

Exercises 10

  1. Which of the following sets is a Dedekind cut?
  2. What property must a set \( A \) have to be a cut?
  3. Which real number does the cut of negative rationals or rationals with square less than \( 4 \) represent?
  4. Which set of rationals defines the cut of \( \sqrt{3} \)?
  5. A Dedekind cut \( A \) must differ from \( \mathbb{Q} \). Why?
  6. Let \( A=\{q\in\mathbb{Q}: q<0 \text{ or } q^2<5\} \). Which real does \( A \) represent?
  7. What is the supremum in \( \mathbb{R} \) of the cut \( A=\{q\in\mathbb{Q}: q<1/2\} \)?
  8. If \( A \) and \( B \) are cuts, what is the cut of \( A+B \) in terms of \( A \) and \( B \)?
  9. Why is \( \mathbb{Q} \) not complete while \( \mathbb{R} \) is?
  10. Let \( A=\{q: q<0 \text{ or } q^2<2\} \) and \( B=\{q: q<0 \text{ or } q^2<2\} \). What is \( A\cdot B \) as a real number, in the positive case?

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.