Axiomatic Emergence B; The Physical Origin of Set Theory

This paper, based on two fundamental physical axioms—Information Conservation and Finite Simulability—demonstrates that the structures of set theory inmathematics can naturally emerge from physical principles. The study shows thatin a closed, self-referential computational system satisfying these two axioms, thehistory of its state evolution is inevitably organized into a binary tree structure offinite depth. Nodes in the tree form equivalence classes based on the indistinguishability of macroscopic physical observations, and their number decays exponentiallywith evolution depth. By establishing a recursive mapping on this tree structure,we prove that its partial order relations precisely satisfy all axioms of HereditarilyFinite Set Theory (ZF¬Infinity), while the Axiom of Infinity does not hold in thisfinite universe. Furthermore, this discrete finite theoretical framework leads to afalsifiable cosmological prediction: a characteristic cutoff should appear in the largescale structure power spectrum near the wavenumber k ≈ 1Mpc−1, which providesa pathway for testing the theory via next-generation observational facilities (suchas SKA-2).