Every system of thought begins with assumptions. The deepest assumption in Western intellectual history — older than Euclid, older than Aristotle — is that the boundless exists. That there is no ceiling. That reality, or at least the mathematics that describes it, extends without limit. This assumption was not argued for. It was inherited, formalized, and forgotten. This programme examines it directly, finds it unforced, and constructs a complete alternative: a foundation for mathematics, physics, and thought built from the ground up on finite bounds.
These are not two independent claims. The second is the only logical consequence of the first. When infinity is genuinely rejected — not relocated, not repackaged — there must be a ceiling. The bound is what makes the rejection real.
The programme proceeds from ontology through logic, set theory, and number systems to the full apparatus of working mathematics and its physical applications.
The complete programme. Fourteen parts: from the foundational package through real analysis, complex analysis, functional analysis, representation theory, complexity theory, and the Millennium Problems.
Every mathematical tool that basic physics requires traced to specific constructions in BST. Experimental grounding across nine areas — from planetary orbits through lattice QCD hadron masses.
75+ named paradoxes from mathematics, logic, physics, and philosophy — each stated, diagnosed, and classified. Four mechanisms account for the vast majority. Seven survive.
The ontological case. The forced-move argument. The parsimony argument. The paradox dividend. The ceiling resolution. No formalism — pure ontology.
The logical substrate. Every quantifier bounded. Complete metatheory: soundness, completeness, decidability, cut-elimination, Craig interpolation, Beth definability.
One axiom. Nine ZFC properties as theorems. Every model finite. The interior/ceiling partition. The Burali-Forti resolution. Consistency relative to IΣ₁.
The complete number chain ℕ_B ↪ ℤ_B ↪ ℚ_B ↪ ℝ_B ↪ ℂ_B. Exact arithmetic on discrete systems. Precision-parameterised identities on continuous systems.
Bounded induction and bounded recursion — the proof engine and computational engine of finite mathematics. Addresses self-grounding (Theorem 8.1) — application to the wider programme pending.
The programme is designed to be entered from multiple directions depending on your background and interest.
The forced-move argument, the parsimony case, the paradox dividend. No formalism required. Follow the consequences into the formal papers if the argument is persuasive.
BFOL's syntax, semantics, proof theory, and metatheory. One change to FOL — bounded quantifiers as primitive — with substantial consequences. Then see what gets built on top.
One axiom. Nine ZFC properties as theorems. Every model finite. The interior/ceiling partition. The Burali-Forti resolution. A different theory, not a weaker one.
ℕ_B through ℂ_B constructed explicitly. Exact arithmetic on discrete systems, precision-parameterised identities on continuous systems. Then the full analytic apparatus in the main paper.
Every mathematical tool basic physics requires, traced to bounded constructions. Experimental grounding across nine areas. Then work backward into the foundations.
"The assumption did not become more justified by becoming more precise. It became harder to question." — The Axiom of Finite Bounds, Working Paper 2026