Physics Beyond The Set‐Theoretic Axiom of Foundation

Classical mechanics implicitly use Archimedes’ axiom, according to that everything may be measured by a rigid scale. Uncertainty in quantum mechanics shows the limits of applying Archimedes’ axiom. The negation of Archimedes’ axiom is derivative from the negation of the set‐theoretic axiom of foundation. The latter postulates that the set‐membership relation is well‐founded: for every set there exists no infinitely descending chain. Denying the foundation axiom in number systems implies setting a non‐Archimedean ordering structure. The main claim of our paper is that physical reality may be regarded as non‐well‐founded in the framework of probabilities distributed on non‐Archimedean ordering structures, in particular, distributed on p‐adic numbers.