German mathematician Richard Dedekind published his construction of real numbers in 1872 in "Continuity and Irrational Numbers" (Stetigkeit und irrationale Zahlen). His insight was that every point on a line corresponds to a unique "cut" that separates the rationals into two parts.
"I find the essence of continuity... in the following principle: If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division."
— Richard Dedekind, 1872
Three equivalent approaches to defining real numbers
Richard Dedekind (1872)
A real number is defined as a partition of ℚ into two sets (A, B) where every element of A is less than every element of B
Advantage: Intuitive geometric interpretation as 'cutting' the number line
√2 is the cut where A = {r ∈ ℚ : r < 0 or r² < 2}
Georg Cantor (1872)
A real number is an equivalence class of Cauchy sequences of rationals
Advantage: Natural connection to limits and convergence
√2 = [(1, 1.4, 1.41, 1.414, ...)]
Various (historical)
A real number is an infinite decimal expansion
Advantage: Familiar and computational
π = 3.14159265358979...
Why we need to go beyond the rational numbers
Although rational numbers are dense (between any two rationals, there's another rational), the rational line has "holes" — points that should exist but don't correspond to any fraction.
Proof by contradiction:
This means there's a "gap" on the rational number line where √2 should be — a length exists (the diagonal of a unit square) that no rational can measure exactly.
A rigorous definition of real numbers
A Dedekind cut is a partition of the rational numbers ℚ into two non-empty sets (A, B) such that:
Every rational belongs to exactly one of A or B
Both sets must contain rationals
Every element of A is less than every element of B
A has no greatest element
Condition D4 is crucial: it prevents A from having a "last element," which would trivially identify the cut. By requiring A to be "open" on the right, cuts can represent both rationals (where B has a minimum) and irrationals (where B has no minimum).
When B has a minimum element r, the cut represents the rational number r.
Example: Cut for 3
Here B has minimum 3, so this cut represents 3.
When B has no minimum element, the cut represents an irrational number.
Example: Cut for √2
B has no minimum — there's always a smaller rational with square > 2.
The complete axiomatic foundation of the real numbers
Closure
Sum of reals is real
Commutativity
Order doesn't matter
Associativity
Grouping doesn't matter
Identity
Zero is additive identity
Inverse
Every number has an additive inverse
Closure
Product of reals is real
Commutativity
Order doesn't matter
Associativity
Grouping doesn't matter
Identity
One is multiplicative identity
Inverse
Non-zero numbers have reciprocals
Distribution
Multiplication distributes over addition
Any two numbers are comparable
Order is transitive
Adding same number preserves order
Multiplying by positive preserves order
Every non-empty subset of ℝ that is bounded above has a least upper bound (supremum) in ℝ.
In ℝ, the set {x : x² < 2} has supremum √2.
In ℚ, the same set has no supremum (√2 ∉ ℚ).
The completeness axiom is what makes calculus work. It guarantees that limits exist, that continuous functions on closed intervals attain their maximum and minimum, and that the Intermediate Value Theorem holds. Without it, analysis falls apart.
For every Dedekind cut (A, B) of the real numbers, the set B has a minimum element. In other words, the real numbers have no "gaps."
Given: A Dedekind cut (A, B) of ℝ.
To show: B has a minimum element.
1. A is non-empty and bounded above (by any element of B).
2. By the completeness axiom (Least Upper Bound Property), exists in ℝ.
3. We claim α ∈ B and α = min B.
4. α ∉ A: If α ∈ A, then since A has no maximum, ∃ a' ∈ A with a' > α. But then α is not an upper bound of A. Contradiction.
5. α = min B: Since A ∪ B = ℝ and α ∉ A, we have α ∈ B. If b ∈ B and b < α, then b is an upper bound of A smaller than sup A. Contradiction.
Problem:
Define the Dedekind cut that represents √2
Solution:
Verify A ∪ B = ℚ: Every rational is in exactly one set
Verify A has no maximum: For any r ∈ A with r > 0, we can find r' > r with (r')² < 2
Verify B has no minimum: For any r ∈ B, we can find r' < r with (r')² > 2
This cut represents √2 because it 'cuts' ℚ exactly where √2 would be
Problem:
Show that the rationals fail the completeness axiom
Solution:
S is non-empty (e.g., 1 ∈ S since 1² = 1 < 2)
S is bounded above (e.g., 2 is an upper bound since 2² = 4 > 2)
Suppose sup S = p/q exists in ℚ
Then either p²/q² < 2, p²/q² = 2, or p²/q² > 2
Case p²/q² < 2: p/q ∈ S, but we can find larger rationals in S (contradiction)
Case p²/q² = 2: Impossible since √2 is irrational
Case p²/q² > 2: p/q is not the least upper bound (contradiction)
Therefore, S has no supremum in ℚ, so ℚ is not complete
Problem:
Express 1/3 as a decimal and explain why it's infinite
Solution:
Perform long division: 1 ÷ 3 = 0.333...
After each step: 10 ÷ 3 = 3 remainder 1
The remainder always returns to 1, creating a cycle
This is an infinite but periodic decimal
Key insight: All rationals have eventually periodic decimals
Problem:
Define addition for the Dedekind cuts representing 2 and 3, and verify the result represents 5
Solution:
Cut for 2: A₂ = {r ∈ ℚ : r < 2}, B₂ = {r ∈ ℚ : r ≥ 2}
Cut for 3: A₃ = {r ∈ ℚ : r < 3}, B₃ = {r ∈ ℚ : r ≥ 3}
Define A₂ + A₃ = {a + b : a ∈ A₂, b ∈ A₃}
For any r₁ < 2 and r₂ < 3, we have r₁ + r₂ < 5
For any r < 5, we can write r = r₁ + r₂ with r₁ < 2, r₂ < 3
Therefore A₂ + A₃ = {r ∈ ℚ : r < 5}, which is exactly the cut for 5 ∎
Problem:
Prove that between any two real numbers a < b, there exists a rational number
Solution:
Given a < b, let h = b - a > 0
By Archimedean property, ∃n ∈ ℕ with 1/n < h
Consider the set S = {m ∈ ℤ : m/n > a}
S is non-empty (by Archimedean) and bounded below
Let m₀ = min S, so m₀/n > a and (m₀-1)/n ≤ a
Then m₀/n ≤ a + 1/n < a + h = b
Therefore a < m₀/n < b, and m₀/n ∈ ℚ ∎
Problem:
Prove the Archimedean property follows from completeness
Solution:
Suppose the Archimedean property fails
Then ∃x > 0 such that ∀n ∈ ℕ, n ≤ x
This means ℕ is bounded above by x
By completeness, ℕ has a supremum α = sup ℕ
Since α-1 < α, α-1 is not an upper bound of ℕ
So ∃m ∈ ℕ with m > α-1, meaning m+1 > α
But m+1 ∈ ℕ, contradicting α = sup ℕ
Therefore the Archimedean property must hold ∎
Problem:
Prove that each real number corresponds to a unique Dedekind cut
Solution:
Suppose cuts (A₁, B₁) and (A₂, B₂) represent the same real α
If A₁ ≠ A₂, WLOG assume ∃r ∈ A₁ with r ∉ A₂
Then r ∈ B₂, so r ≥ α (since α separates A₂ and B₂)
But r ∈ A₁ means r < α (since α separates A₁ and B₁)
This is a contradiction, so A₁ = A₂
Similarly B₁ = B₂, proving uniqueness ∎
Problem:
Show that between any two distinct rationals, there exists an irrational
Solution:
Let p, q ∈ ℚ with p < q
Consider r = p + (q-p)/√2
Since √2 is irrational, (q-p)/√2 is irrational
Sum of rational p and irrational (q-p)/√2 is irrational
Also p < r < q since 0 < (q-p)/√2 < (q-p)
Therefore r is irrational and p < r < q ∎
Fundamental results about real numbers
Every Dedekind cut of the rationals determines a unique real number.
Proof Sketch: By construction, each cut IS a real number. The uniqueness follows from the definition of equality between cuts.
Between any two distinct real numbers, there exists a rational number.
Proof Sketch: Given a < b, by Archimedean property find n with 1/n < b - a. Then some multiple m/n lies in (a, b).
Between any two distinct real numbers, there exists an irrational number.
Proof Sketch: Given a < b, the number a + (b-a)/√2 is irrational and lies strictly between a and b.
If [aₙ, bₙ] is a sequence of nested closed intervals with lengths → 0, then ∩[aₙ, bₙ] contains exactly one point.
Proof Sketch: The sequences {aₙ} and {bₙ} are bounded and monotonic, so they converge. Since bₙ - aₙ → 0, they converge to the same limit.
An alternative to Dedekind cuts using sequences
A Cauchy sequence is a sequence where terms get arbitrarily close to each other. We define real numbers as equivalence classes of Cauchy sequences of rationals, where two sequences are equivalent if their difference converges to 0.
Between any two reals lies a rational
For any real numbers a < b, there exists a rational r with a < r < b. This follows from the Archimedean property. Despite being 'dense', rationals still have 'gaps' that irrationals fill.
Cantor's diagonal argument
The real numbers are uncountable — there are 'more' reals than rationals. This is proven by Cantor's diagonal argument, showing any attempted list of reals must miss some numbers.
Two types of irrational numbers
Algebraic numbers (like √2) are roots of polynomials with integer coefficients. Transcendental numbers (like π and e) are not. Surprisingly, 'almost all' real numbers are transcendental.
Introduced Dedekind cuts in 1872 to give a rigorous definition of real numbers
His construction showed how to 'fill the gaps' in the rationals to create the continuum
Developed the Cauchy sequence construction of reals and proved their uncountability
Showed that there are 'more' reals than rationals, revolutionizing our understanding of infinity
Developed the concept of Cauchy sequences and convergence criteria
His sequences provide an alternative construction of real numbers equivalent to Dedekind cuts
Real number completeness ensures smooth curves and continuous color gradients
Example: Bezier curves rely on the completeness axiom for guaranteed intersection points
Continuous motion and fields require the completeness of real numbers
Example: The Intermediate Value Theorem guarantees that a ball thrown upward reaches every height in between
Equilibrium existence proofs use the completeness of real numbers
Example: Brouwer's Fixed Point Theorem (based on completeness) proves Nash equilibria exist
Convergence of algorithms relies on completeness
Example: Newton's method converges because bounded monotonic sequences converge in ℝ
Decimals are a representation, not a definition. To prove theorems about real numbers (like the Intermediate Value Theorem), we need a rigorous foundation. Dedekind cuts provide this by defining reals as 'cuts' in the rationals, making completeness provable rather than assumed.
The Dedekind cut for √2 has no maximum in its lower set and no minimum in its upper set. This means no rational number 'sits at the cut' — the cut falls in a 'gap' between rationals, which is precisely what we mean by an irrational number.
The completeness axiom cannot be proven from the field and order axioms alone — it must be added separately. The rationals satisfy all field and order axioms but fail completeness. This axiom is what 'fills the gaps' and makes calculus work.
No! Dedekind cuts are one approach. Others include Cauchy sequences (defining reals as equivalence classes of convergent rational sequences) and the axiomatic approach (just listing the axioms). All methods yield isomorphic structures.
Using the completeness axiom: Let x = 0.999... The Dedekind cut for x has the same lower set as the cut for 1 (all rationals less than 1). Since cuts are determined by their lower sets, x = 1. Alternatively: 1 - 0.999... would be a positive real smaller than all 1/10ⁿ, which can't exist.
The Archimedean property states that for any real numbers x > 0 and y, there exists a natural number n such that nx > y. This means there are no 'infinitely large' or 'infinitely small' real numbers. It follows from completeness and is crucial for limits.
To add cuts (A, B) and (C, D), we form (A + C, B + D) where A + C = {a + c : a ∈ A, c ∈ C}. This corresponds exactly to adding the real numbers they represent. Multiplication is defined similarly but requires more care with signs.
They are equivalent! The Nested Interval Property states that any sequence of nested closed intervals [aₙ, bₙ] with lengths → 0 has exactly one point in common. This can be proven from completeness, and vice versa.
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