A Constructive Fragment of Physical Propositions

TL;DR

This paper analyzes the limits of measurement in Minkowski spacetime, showing that only finite observational sequences are meaningful and establishing an operational form of incompleteness related to the evidential constraints of measurement.

Contribution

It introduces a proof-theoretic framework linking measurement admissibility to arithmetical fragments and demonstrates an operational incompleteness theorem based on measurement limitations.

Findings

Admissible measurement yields only finite observational sequences.

Physically meaningful propositions correspond to specific arithmetical fragments.

There exist true arithmetical propositions about measurements that no recursive theory can decide.

Abstract

We develop a proof-theoretic analysis of the Operational Standard of Matsas, Pleitez, Saa, Vanzella (2024) showing that admissible measurement in Minkowski Spacetime yields only finite observational sequences and thereby restricts the class of physically meaningful propositions to those admitting terminating extraction procedures or uniform stability conditions. These correspond exactly to the arithmetical fragment $\Sigma^0_1 \cup \Pi^0_2$, and the induced realizability structure interprets $\Delta_0$ Heyting Arithmetic on the code of observational data. A diagonal argument then establishes an operational form of incompleteness: there exist true arithmetical propositions about admissible extraction that no sound, recursively axiomatizable theory of spacetime can decide. The result is structurally analogous to classical incompleteness but arises from the evidential limits of measurement rather than from ontological assumptions.