The complete number chain ℕ_B(k) ↪ ℤ_B(k) ↪ ℚ_B(k²) ↪ ℝ_B(k) ↪ ℂ_B(k⁴), constructed from Bounded Set Theory alone. Exact arithmetic on discrete systems. Precision-parameterised identities on continuous systems. Every object finite, every cardinality bounded, every construction verified through the nine BFTs. The Cayley-Dickson extensions to bounded quaternions and octonions are available at the algebraic level.
Discrete systems (ℕ, ℤ, ℚ) have exact arithmetic on interior elements. Continuous systems (ℝ, ℂ) have rational computation followed by rounding, with grid displacement bounded by O(1/k²) per operation.
Von Neumann ordinals {0, 1, ..., k}. Arithmetic by bounded recursion. Unique factorisation, GCD, Bezout, primality — all by BI-BST with finite search.
Pairs of naturals modulo (a,b) ~ (c,d) iff a+d = b+c. Equivalence checked in ℕ_B(2k). Ring axioms on the interior domain. Integers from −k to +k.
Fraction pairs modulo ad = bc, checked in ℤ_B(k²). Field axioms on the interior domain. k²-dense: no gap exceeds 1/k² (Farey sequence argument).
Cauchy sequences rounded to nearest rational via ρ_k. Equivalence by exact identity ρ_k(s) = ρ_k(t). k-complete: every Cauchy sequence has a limit. Commutativity exact; associativity at O(1/k²).
ℝ_B(k) × ℝ_B(k). Standard complex arithmetic with rounding. i² = (−1, 0) exact. Non-orderable. Algebraic closure for degrees 1–4 by explicit formulas.
At each quotient stage of the chain, equivalence is defined by exact identification computed within the model's resources. If any equivalence were approximate, the embeddings would fail. Exact equivalence is a structural requirement, not a convenience.
At each quotient stage, equivalence is defined by exact identification computed within the model's resources. The pattern is uniform: exact equality, trivially transitive, no approximate threshold. This is forced by BST's requirement that all constructions use decidable conditions on interior elements.
Each construction consumes a bounded number of rank levels. The super-exponential growth of the cumulative hierarchy (1, 2, 4, 16, 65536, ...) ensures room for the next construction. For any k, 𝒱_{k⁴+8} contains the entire chain with all components interior.
On discrete systems, arithmetic is exact for interior results. On continuous systems, every operation produces a definite value and the grid displacement is bounded: ≤ 1/(2k²) per operation for associativity, ≤ C/k² for distributivity. The precision parameter is the mathematical content.
Every major result from the number chain is classified. Most are Type I.
No precision parameters, no family indexing. The theorem is simply true within the bounded system about its own objects.
BST proves the theorem at each k with stable form across the family. The classical theorem is the schema the instances share.
BST proves the theorem at each k with an explicit precision parameter quantifying the grid displacement. The parameter is the mathematical content — it reports what computation actually produces.
The Cayley-Dickson construction iterates the Cartesian product with modified multiplication. At each step, an algebraic property is lost. The loss is a feature of the construction, not of the bounded setting.
Every object finite, every cardinality bounded, every construction verified. The analytical apparatus over ℝ_B(k) and ℂ_B(k⁴) — continuity, differentiation, integration, convergence, transcendental functions — belongs to the main paper, The Axiom of Finite Bounds.
"Every verified proof in mathematics is a finite derivation over finite symbols. BST makes this finiteness explicit rather than suppressing it behind an infinite ontology." — Bounded Number Theory, Working Paper 2026