[Paper Review] Building Cantor's Bijection

Samuel Nicolay, Laurent Simons|arXiv (Cornell University)|Aug 28, 2014

semigroups and automata theory6 references3 citations

TL;DR

This paper constructs a bijection between the unit square [0,1]² and the unit interval [0,1] by modifying Cantor's original non-surjective decimal-based mapping using the Schröder-Bernstein theorem. Starting from Cantor's function 𝒞 that fails to be surjective due to non-unique decimal expansions, the authors define a new bijection f* that agrees with 𝒞 almost everywhere but corrects its deficiencies on a null set via iterative injection-based refinement, yielding a fully bijective correspondence between the square and the interval.

ABSTRACT

Cantor's first idea to build a one-to-one mapping from the unit interval to the unit square did not work since, as pointed out by Dedekind, the so-obtained function is not surjective. Here, we start from this function and modify it (on a negligible set) in order to obtain the desired result: a one-to-one correspondance between the unit interval and the unit square.

Motivation & Objective

• To resolve Cantor's original failed attempt to construct a bijection between [0,1]² and [0,1] using decimal interleaving.
• To address Dedekind's critique that Cantor's decimal-based function 𝒞 is not surjective due to non-proper decimal expansions.
• To construct a fully bijective mapping from the unit square to the unit interval using the Schröder-Bernstein theorem.
• To provide a constructive, explicit bijection that corrects Cantor's initial function on a negligible set while preserving its structure almost everywhere.

Proposed method

• Define an injection f: [0,1]² → [0,1] based on interleaving proper decimal expansions of x and y, with special handling for boundary points (x=1 or y=1).
• Define a reverse injection g: [0,1] → [0,1]² via g(t) = (t, 0), ensuring injectivity.
• Apply the Schröder-Bernstein theorem by constructing sequences (Aₙ) and (Bₙ) of subsets of [0,1]² and [0,1] respectively, starting from A₀ = [0,1]×(0,1] and B₀ = f(A₀).
• Iteratively define Aₙ = g(Bₙ₋₁) and Bₙ = f(Aₙ) for n ≥ 1, capturing the image chains of the injections.
• Construct the final bijection f* by setting f*(x,y) = f(x,y) if (x,y) ∈ ⋃ₙ Aₙ, and f*(x,y) = x otherwise, ensuring bijectivity.
• Prove that the correction set, where f* differs from f, lies within [0,1]×{0}, a Lebesgue-negligible set in ℝ².

Experimental results

Research questions

• RQ1 Can Cantor’s original decimal-based interleaving function be modified to yield a true bijection between [0,1]² and [0,1]?
• RQ2 Why does Cantor’s initial function fail to be surjective, and how can this failure be systematically corrected?
• RQ3 Can the Schröder-Bernstein theorem be applied constructively to build an explicit bijection from a pair of injections, even when one injection is not surjective?
• RQ4 What is the measure-theoretic size of the set where the corrected bijection differs from the original decimal interleaving function?
• RQ5 How can the structure of the correction set be explicitly characterized in terms of decimal expansion patterns?

Key findings

• The function f* defined by f*(x,y) = f(x,y) if (x,y) ∈ ⋃ₙ Aₙ and f*(x,y) = x otherwise is a well-defined bijection from [0,1]² to [0,1].
• The set where f* differs from the original function 𝒞 is contained in [0,1]×{0}, which has Lebesgue measure zero in ℝ², so f* = 𝒞 almost everywhere.
• The construction explicitly identifies the correction set as the set of points (x,y) not in any Aₙ, which are precisely those with y=0 and x not in the image of any Bₙ under g.
• The sequences (Aₙ) and (Bₙ) are strictly defined via iterative application of f and g, with Aₙ = g(Bₙ₋₁) and Bₙ = f(Aₙ), forming a descending chain of sets.
• The bijection f* is injective and surjective because it combines the bijective image of f on ⋃ₙ Aₙ with the bijective image of g⁻¹ on the complement, satisfying the conditions of the Schröder-Bernstein theorem.
• The final mapping f* preserves the intuitive structure of Cantor’s decimal interleaving while resolving its non-surjectivity through a minimal, well-characterized correction on a null set.