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(FAPP) Infinity Does Macroscopic Irreversibility From Microscopic Reversibility

Karl Svozil

TL;DR

Infinity is essential to deriving macroscopic irreversibility from microscopic reversibility, bridging rational constructions of $\mathbb{R}$ and physical theories via infinite processes, including Specker sequences and Chaitin's $\Omega$. The paper argues that equivalence relations capture For-All-Practical-Purposes indistinguishability across classical analysis, quantum sectorization via infinite tensor products, and statistical macrostates, enabling emergent irreversibility. It develops sectorization and non-unitary equivalence in infinite tensor product spaces as a route to reconcile unitary evolution with measurement-induced apparent collapse, and formalizes FAPPness using equivalence classes in three domains. This unifies mathematical, computational, and physical perspectives on how infinity and coarse-graining yield robust macroscopic behavior from reversible microdynamics, with implications for measurement, thermodynamics, and the foundations of physics.

Abstract

Infinity is central to deriving macroscopic irreversibility from reversible microscopic laws across mathematics, theoretical computer science and physics. In analysis, infinite processes - such as Dedekind cuts and Cauchy sequences - construct real numbers as equivalence classes of rational approximations, bridging discrete rationals to the continuous real line. In quantum mechanics, infinite tensor products model nested measurements, where sectorization partitions the Hilbert space into equivalence classes, reconciling unitary evolution with wavefunction collapse. In statistical mechanics, macrostates emerge as equivalence classes of microstates sharing identical macroscopic properties, providing the statistical basis for thermodynamic irreversibility despite reversible dynamics. Equivalence relations formalize For-All-Practical-Purposes (FAPP) indistinguishability, reflecting operational limits on precision and observation. Together, these examples reveal a unified framework where infinity and equivalence underpin emergent macroscopic behavior from microscopic reversibility.

(FAPP) Infinity Does Macroscopic Irreversibility From Microscopic Reversibility

TL;DR

Infinity is essential to deriving macroscopic irreversibility from microscopic reversibility, bridging rational constructions of and physical theories via infinite processes, including Specker sequences and Chaitin's . The paper argues that equivalence relations capture For-All-Practical-Purposes indistinguishability across classical analysis, quantum sectorization via infinite tensor products, and statistical macrostates, enabling emergent irreversibility. It develops sectorization and non-unitary equivalence in infinite tensor product spaces as a route to reconcile unitary evolution with measurement-induced apparent collapse, and formalizes FAPPness using equivalence classes in three domains. This unifies mathematical, computational, and physical perspectives on how infinity and coarse-graining yield robust macroscopic behavior from reversible microdynamics, with implications for measurement, thermodynamics, and the foundations of physics.

Abstract

Infinity is central to deriving macroscopic irreversibility from reversible microscopic laws across mathematics, theoretical computer science and physics. In analysis, infinite processes - such as Dedekind cuts and Cauchy sequences - construct real numbers as equivalence classes of rational approximations, bridging discrete rationals to the continuous real line. In quantum mechanics, infinite tensor products model nested measurements, where sectorization partitions the Hilbert space into equivalence classes, reconciling unitary evolution with wavefunction collapse. In statistical mechanics, macrostates emerge as equivalence classes of microstates sharing identical macroscopic properties, providing the statistical basis for thermodynamic irreversibility despite reversible dynamics. Equivalence relations formalize For-All-Practical-Purposes (FAPP) indistinguishability, reflecting operational limits on precision and observation. Together, these examples reveal a unified framework where infinity and equivalence underpin emergent macroscopic behavior from microscopic reversibility.

Paper Structure

This paper contains 23 sections, 12 equations, 1 figure.

Table of Contents

  1. From Rationals to Reals: The Role of Infinity
  2. Dedekind Cuts
  3. Cauchy Sequences
  4. Infinite Decimal Expansions
  5. Cantor's Diagonalization and Irrational Numbers
  6. No Continua Without Infinite Means
  7. Specker Sequences and the Role of Infinity
  8. Chaitin's Omega as the Ultimate Specker Sequence
  9. Infinite Tensor Products and the Quantum Measurement Problem
  10. Infinite Tensor Products in Nested Measurement Scenarios
  11. The Von Neumann-Landau Measurement Scheme
  12. Challenges with Infinite Tensor Products
  13. Cardinality
  14. Inner Product and Orthogonality
  15. Sectorization as a Solution
  16. ...and 8 more sections

Figures (1)

  • Figure 1: Evolution of the number of balls in Urn 1 (blue) and Urn 2 (dotted, orange) in the Ehrenfest Urn Model (N=100, 1000 steps), starting from a low-entropy state with Urn 1 filled and Urn 2 empty. The dashed line marks the equilibrium state (N/2=50). The simulation highlights the system's relaxation towards equilibrium and the persistent fluctuations around it, illustrating the microscopic reversibility that underlies Zermelo's recurrence objection.