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

Gödel’s Second Incompleteness Theorem establishes that no sufficiently strong, consistent formal system can prove its own consistency, apparently placing absolute mathematical certainty beyond the reach of formal methods. Yet Gödel’s philosophical notebooks — the Max Phil — reveal an unwavering commitment to mathematical realism: the conviction that mathematics describes an objective, mind-independent reality. We argue that these two pillars of Gödel’s legacy are not in tension but mutually illuminating, pointing toward a single resolution that Gödel himself sought. The external “meta-system” required to ground arithmetic’s consistency is the physical world itself. Our argument pivots on the Model Existence Theorem: a formal system is consistent if and only if it has a model. We propose that the physical universe — understood as an ontological substrate whose description via Quantum Field Theory (QFT) functions as an Effective Field Theory (EFT) — constitutes a non-circular physical model of the Peano axioms. The non-circularity rests on the Tarskian distinction between a model and its metalanguage: QFT presupposes arithmetic in its formulation, but the physical states it describes exist independently of that formulation. The ontological commitment to those states is grounded in ontic structural realism (OSR) — adopted here as the most parsimonious framework for keeping physical structure categorically distinct from its mathematical representation, and compatible with a range of broader ontological positions — and in the criterion of stability under renormalization group (RG) flow. We locate this proposal in dialogue with the mathematical universe hypothesis of Tegmark and the formal-limit approach of Kiefer, arguing that our inversion of the mathematics-physics dependence relation offers a more satisfying resolution of Gödel’s foundational program. The consistency of arithmetic is not an article of faith, but a consequence of the coherent existence of the cosmos. This conclusion stands in deliberate contrast to Tegmark’s Mathematical Universe Hypothesis, which runs the dependence relation in the opposite direction: where Tegmark grounds physical existence in mathematical structure, we argue that physical existence grounds mathematical consistency.