Constructivist vs Mainstream Real Number Lab

Mathematical Framework

Bishop Constructivism: A real number is valid only if an explicit algorithm provides rational approximations with a known convergence modulus. Non-constructive existence proofs and uncomputed limits are rejected.

Select Real Number or Sequence

Target Precision Decimal Places (k)

Algorithmic Step Budget

250

Foundational Definition

x = (x_n) with |x_n - x_m| ≤ n⁻¹ + m⁻¹

Errett Bishop (1967): A real number is a sequence of rationals with an explicit, constructive Cauchy modulus. One cannot claim the existence of a completed infinite decimal without an algorithm to compute every k-th digit in finite steps.

Decimal Expansion Stream Constructive Mode

Specker Sequence / Uncomputable Real Approximation Bound: ±10⁻⁶

Algorithmic Cauchy Modulus Bounds Interactive Visualization

Constructive Interval Analysis: Each rational term xₖ guarantees x ∈ [xₖ - 1/k, xₖ + 1/k]. In the constructivist worldview, an infinite decimal is never an arbitrary completed object; it is an active procedure generating shrinking rational intervals.

Constructive vs Classical Ledger Verified State

Mainstream (Platonist)

Accepts Law of Excluded Middle (LEM). Number exists definitively as a Dedekind cut. All digits exist simultaneously in Platonic realm regardless of whether a Turing machine can halt to find them.

Constructivist (Bishop)

Rejects LEM for uncompleted infinite tests. A digit is undefined until its constructive witness is found in finite operations. Uncomputable digits remain algorithmically bounded, not completed.

Runtime Telemetry

Mode	constructive
Target Digits (k)	6
Step Budget	250 ops
Convergence Bound	≤ 2.000000e-6
LEM Invocation	Rejected (Bishop)
Halting Proof	Required / Incomplete
Dedekind Cut Decision	Weakly Undecidable

JSON Research Export

Canonical mathematical report with sequence digits, constructive validation state, and foundational differences.