The ontological case for bounded finitude. Not a technical restriction on an existing framework — a claim about what exists. The forced-move argument shows that rejecting infinity without asserting a bound does not produce a finite mathematics. It produces an infinite mathematics in which every individual object happens to be finite. The bound is what makes the rejection real.
These are not two independent claims. The second is the only logical conclusion of the first. When infinity is genuinely rejected — not relocated from objects to domains, not repackaged as an endless process, not redistributed from sets to the theory that describes them — there must be a ceiling. A maximum. An upper bound on what exists. The bound is what makes the rejection real rather than verbal.
When you deny completed infinity, exactly three positions are available. Two of them fail to deliver what they promise. They do not eliminate infinity — they relocate it.
Every collection is finite, but for every finite collection there is a strictly larger one. The process of generating larger finite sets never terminates. This is the Aristotelian position, refined by Brouwer. A process that never terminates is an infinite process. A domain that contains finite collections of every size contains infinitely many objects. The infinity has not been eliminated. It has been relocated from the objects to the process.
Every set is finite, and that is all. No claim about a process, no talk of potentiality. This is ZF¬∞. But for every natural number n, ZF¬∞ proves the existence of a set with n elements — without limit. Any model satisfying all these sentences simultaneously contains infinitely many sets. Every set is finite; the universe of sets is infinite.
Every set is finite, and there is a ceiling — an upper bound on the size of any set that exists. The bound is finite and real, even if its value is unknown. This is the position of this paper. Only bounded finitude eliminates infinity from both the objects and the domain. The bound is not an eccentric addition. It is the necessary consequence of taking the first claim seriously.
The forced-move argument shows the bound is necessary if infinity is rejected. The parsimony argument goes further: it asks why infinity should be accepted in the first place. The answer is not that infinite mathematics is wrong. It is that infinite mathematics posits more than it needs to.
The complete bounded number chain, real analysis, complex analysis, functional analysis, representation theory, and complexity theory — all constructed within BST. The infinite commitment is unforced. You can have the mathematics without it.
Every experimentally verified prediction in the physical record was computed by finite methods: numerical integration, finite matrix diagonalisation, finite sums of Feynman diagrams, lattice Monte Carlo. The infinite continuum is not load-bearing for physics. It is scaffolding.
BST removes not individual paradoxes but the four mechanisms that generate them. A survey of over seventy known infinity-dependent paradoxes finds that all require at least one mechanism BST eliminates.
The existence of ℕ, ℝ, or any infinite collection as a finished object. Generates Hilbert's Hotel, Galileo's paradox, all supertask paradoxes, and transfinite cardinal arithmetic.
Selection from uncountable collections where no constructive procedure is possible. Generates Banach-Tarski, Vitali non-measurable sets, the well-ordering of ℝ, and pathological decompositions.
The collection of all subsets of an infinite set as a completed set. Generates the hierarchy of uncountable infinities, Cantor's paradox, and the Continuum Hypothesis.
∀x and ∃x ranging over completed infinite totalities. Generates Skolem's paradox, enables Berry's and Richard's paradoxes, and forces countable models of "uncountable" set theory.
Any theory asserting a maximum in a domain closed under construction faces a contradiction. BST resolves this by the same structural move ZFC uses — restricting the scope of construction operations so they do not apply to the maximum object.
The collection of all ordinals is a proper class — real but not representable as a set. Proper classes cannot be members of other classes. Their ontological status is notoriously contested: are they real mathematical objects? A manner of speaking? Sets in a larger universe?
Maximum-cardinality sets are ceiling elements — concrete finite objects within the domain with definite cardinality, but constructively inert. All nine Bounded Fundamental Theorems apply only to interior elements. The bound is carried by real objects, not metaphysically ambiguous entities.
The commitment comes first. The mathematics comes from the commitment.
The logical substrate. Every quantifier carries an explicit bounding term. Unbounded forms are absent from the language. Complete metatheory: soundness, completeness, decidability, cut-elimination, Craig interpolation, Beth definability.
Read more →A complete finite set theory from a single axiom. Nine ZFC axioms proved as Bounded Fundamental Theorems. Every model is finite. Consistency relative to IΣ₁. Proof-theoretic ordinal ω^ω.
Read more →The complete bounded number chain ℕ_B(k) ↪ ℤ_B(k) ↪ ℚ_B(k²) ↪ ℝ_B(k) ↪ ℂ_B(k⁴). Exact arithmetic on discrete systems, precision-parameterised identities on continuous systems. Cayley-Dickson extensions to ℍ_B and 𝕆_B.
Read more →The main paper. Fourteen parts. Analysis, complex analysis, functional analysis, representation theory, complexity theory, and the Millennium Problems — constructed from the ground up on bounded foundations.
Read more →Every mathematical tool that basic physics requires traced to specific constructions in bounded set theory. Experimental grounding across nine areas of physics, from planetary orbits to lattice QCD hadron masses.
Read more →75+ named paradoxes examined. Four mechanisms identified. Four outcomes classified: dissolved, tamed, transformed, preserved. Seven survive — and their survival is structurally informative.
Read more →Thirteen sections. The forced-move argument, the parsimony argument, the paradox dividend, the ceiling resolution, the boundaries stated honestly, and the objections addressed. The commitment stands: there is no infinity, and there is an upper bound. That is not a restriction. It is the form finitude must take.
"This paper does not claim to have proved that infinity does not exist. It demonstrates that a foundation without infinity is coherent, expressive, and sufficient — and it establishes that the burden of justification falls on the party making the greater ontological claim." — Finite Philosophy, Working Paper 2026