The Identity Principle: Finite Foundations for Mathematics and Quantum Mechanics

The Identity Principle: Finite Foundations for Mathematics and Quantum Mechanics proposes a radical reinterpretation of mathematical ontology in a finite physical universe. Under this principle, mathematical entities are not abstract objects but real physical structures — sets are first-order collections of physical objects from the universe $U$, and numbers are defined as the totality of $k$-element configurations.

This physically grounded approach prevents the formation of self-referential sets, thereby resolving Russell’s paradox without requiring axiomatic restrictions. Applied to quantum mechanics, the framework interprets superposition as a description of possible future states $F(t)$ prior to measurement, offering a realist account of wavefunction collapse as the selection of one physical element from that set.

The Identity Principle also rejects actual infinity while retaining calculus and limits through potential infinity. Mathematics thus becomes an internally consistent reflection of physical structure — an ontology of the finite universe itself.

Keywords: finite mathematics, set theory, quantum mechanics, foundations of mathematics, physical ontology, Russell paradox, potential infinity.