The Physical World as a Model of Peano Arithmetic: Inverting the Mathematics–Physics Dependence Relation

Gödel’s Second Incompleteness Theorem shows no sufficiently strong, consistent formal system can prove its own consistency; yet his notebooks reveal a robust mathematical realism. We argue the external ground for arithmetic’s consistency is the physical world. Via the Model Existence Theorem, the universe, described by Quantum Field Theory as an Effective Field Theory, constitutes a model of the Peano axioms. Non-circularity is secured by the Tarskian model/metalanguage distinction; ontological commitment, by structural realism and renormalization-group stability. We invert, without refuting, Tegmark’s Mathematical Universe Hypothesis: physical existence grounds mathematical consistency — not a presuppositionless proof, but a warrant resting on that reversal.