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May 15, 2026Open Access

Locality of Euclid's Theorem

Key Points

The aim is to explore the distinction between procedural unboundedness and actual infinity in Euclid's theorem.
Analyzes the classical proof of prime infinitude and its implications within the Mathematics of a Finite Universe.
Introduces a distinction between procedural generation and the acceptance of infinite sets.
Explores operational limits of Euclid's construction at specific values of p.
Demonstrates that Euclid's theorem, while logically sound, reflects an act of faith regarding infinity.
Shows that the operational realizability of Euclid's procedure diminishes at p = 509 for a finite universe.
Argues that interpreting Euclid's theorem as asserting absolute infinity is not a logical necessity.

Abstract

What if Euclid's theorem is logically flawless, yet ontologically local? This paper argues that the classical proof of the infinitude of prime numbers, while formally correct, does not compel belief in an actually infinite set. Within the Mathematics of a Finite Universe (MFU) — where a finite informational horizon Ωₒnt bounds all physical realizability — Euclid's construction remains a perfect procedural generator, but its interpretation as a description of absolute reality is an act of faith, not a logical necessity. The work introduces a sharp distinction between procedural unboundedness and actual infinity, and shows that Euclid's own procedure loses operational realizability at the surprisingly low boundary of p = 509 (for Ωₒnt = 10²50). A compact contribution to the philosophy of mathematics, physical ontology, and the limits of mathematical realism. Follow my work and related discussions on Facebook: Facebook

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