Construction of the Real Numbers via Dedekind Cuts

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1 · Prerequisites

Set Theory

2 · Logical flowchart

3 · Items

Definition

Construction of the Real Numbers via Dedekind Cuts

3 · Items

The integers as equivalence classes of pairs of naturals

The integer-defining relation is an equivalence relation

Arithmetic on the integers

Integer addition and negation are well defined

Integer multiplication is well defined

The integers form a commutative ring

Order on the integers

The integers form a totally ordered ring

The integers have no zero divisors; multiplicative cancellation

The naturals embed in the integers

The rationals as equivalence classes of pairs of integers

The rational-defining relation is an equivalence relation

Arithmetic on the rationals

Rational arithmetic is well defined

The reciprocal on the rationals is well-defined

The rationals form a field

Order on the rationals

Every rational has a positive-denominator representative

The rationals form a totally ordered field

The integers embed in the rationals

The rationals are Archimedean

Dedekind cut

The real numbers ℝ as Dedekind cuts

Each rational cut q^* is a Dedekind cut

Order on the Dedekind reals

Inclusion totally orders the Dedekind reals

Addition, negation, and subtraction of Dedekind cuts

Cut addition: A+B is a cut, commutative and associative, with identity 0^*

For a cut A, -A is a cut and A + (-A) = 0^*

Multiplication and reciprocals of Dedekind cuts

For a positive cut A, the reciprocal A⁻¹ satisfies A · A⁻¹ = 1^*

The Dedekind reals form a field

The rational cuts embed densely in ℝ, preserving sums, products, 0, 1 and the order

The Dedekind reals form a totally ordered field

The Dedekind reals are Archimedean

Dedekind completeness: the least-upper-bound property

The cut S = {q : q<0 or q²<2} is an irrational real number

FALSE: some rational number squares to 2

FALSE: every Dedekind cut has a greatest element