In 610 BCE, Anaximander declared the fundamental principle of reality to be infinite. No argument established it. No observation confirmed it. For 2,500 years the commitment ran without interruption. Aristotle managed it. Cauchy refined it. Cantor completed it. In 1908 Zermelo wrote it down as a formal axiom, listed alongside Extensionality and Pairing as though it carried the same epistemic weight. But the commitment had never been an axiom in the modern sense. It was an axiom in the ancient sense: a thing assumed, not a thing proved. The formalisation did not justify it. It made it invisible. The problem is not the Axiom of Infinity in ZFC. Remove that axiom and you still have a logic whose quantifiers range over domains of unspecified size, whose compactness theorem forces infinite models, and whose grammar cannot express the sentence "there are only finitely many things." The problem is the logic itself. Every logical framework since Frege permits unbounded quantification. None prevents infinity. Most require it. Bounded Foundations fixes the grammar. Every quantifier carries an explicit bound. Unbounded quantification is not a well-formed formula. From this single syntactic restriction, the book derives bounded set theory (nine classical properties, all proved as theorems), a complete number system from the naturals through the octonions, and the full apparatus of analysis: differentiation, integration, the fundamental theorem of calculus, transcendental functions, Fourier analysis, and differential equations. Every definition is verified in Isabelle/HOL with no axioms assumed and no proofs left incomplete. The bound is where the construction lives, not where it breaks.
